In this article, I will explain the differential equations of fluid motion, i.e., conservation of mass (the continuity equation. So without wasting any time, let us start.
As a part of this article, it is essential to know what differential analysis is and how we can apply it to explain continuity and Naiver Stroke’s theorem.
Some of the essential key points related to the differential analysis are as follows:
Differential analysis is the application of a differential equation of fluid motion to any or every point in the flow field over a region called the Flow Domain.
Some readers might confuse the word differential with the small control volumes piled up on each other in the flow field.
Whenever the size of the control volume crosses the limit and extends to infinity, then the size of each control volume becomes so small that the conservation equations simplify to a set of partial differential equations. These partial equations can easily be applicable whenever required in any flow field.
As I have discussed earlier, two differential equations, the Law of Conservation of Mass (Continuity Equation) and Newton’s Second Law (Naiver Strokes Equation), are the ones on which there is a drastic change in the temperature and density such kind of equations are easily solved with the help of differential equation.
The following diagram shows the study of control volume in which the control volume seems to be much similar to the black box.
Now, the second diagram shows that all the flow points are solved within the flow domain in the case of differential analysis.
While solving the differential equation in the case of incompressible flow, there are about four unknown, i.e., velocity components (u, v, w), one pressure component, and four equations (three equations from Naiver Strokes Law and one from Law of Conservation of Mass).
There are variables and constants in equations, but in differential equations, all the variables are solved at once because, in such situations, the equations are coupled. Now, what is coupled, and how are equations coupled? We will see it in the upcoming topics.
While solving the differential equation, the boundary conditions must be defined.
We are moving towards the first important part of our article, i.e., the Law of Conservation of Mass. Let us start.
In one of my previous articles, I have extensively explained the conservation of mass. But now, here, I will explain the derivation in terms of the infinitesimal control volume by the divergence theorem. So let’s start.
To explain the topic, I will divide it into essential critical points in the following way.
First of all, let us recall the conservation of the mass equation through the application of Reynold’s Transport Theorem;
0=∫CV∂ρ∂t dV+∫CSV.n dA (a)
The equation is for the fixed and control volumes.
In the case of well-defined and selected inlets and outlets, the equation will be as follows:
∫CV∂ρ∂tdV=inm-outn (b)
The above equation explains that the rate of change of mass within the control volume equals the rate at which mass flows into the control volume subtracting the rate with mass flow out of the control volume.
Now I will explain the derivation using the divergence theorem.
Derivation Using the Divergence Theorem:
The other name of the divergence theorem is called Gauss’s Theorem.
The statement of the divergence theorem is as follows:
The Divergence Theorem (Gauss’s Theorem) is used to transform a volume integral of the divergence of a vector into an area integral over the surface that defines the volume.
Some of the essential key points related to the Divergence theorem are as follows:
Mathematically, the divergence theorem is defined as the divergence G and can be written as follows:
∫vV.GdV=∮AG.n dA (c)
There are two kinds of integration in the equation one is simple, and the other has circled it. This indicates that the entire area surrounds the volume. So as you can see that the equation is beneficial in gaining data.
The primary purpose of using the divergence theorem is to transform the volume integral of the divergence vector into an area integral over the surface, and that surface defines the volume.
In the case of any vector, the divergence will be defined as a G, and the equation as m will be used to describe it.
In some cases, we also define the divergence as follows:
G=ρV
We can also define it by playing with the values, and for that, we substitute the value of equation (c) into equation (a), and we will get the results as follows:
0=∫cv∂ρ∂t dV+∫cvV. (ρV) dV
As you can see that there are two integrals, so to get the required equation, we will combine the two integrals into one and then the result will be as follows:
∫cv∂ρ∂t+V. VdV=0
Now, we come to the conclusion that the equation that is mentioned above is for the control volumes, regardless of any size and shape.
So, the statement above is only possible if the terms within the brackets are identically zero.
Moving towards the critical statement, i.e. the equation of continuity. So the general differential equation for the conservation of mass is also known as the continuity equation, and the equation is as follows:
∂ρ∂t+V. V=0
So, this is the equation of continuity. This equation is for compressible flows only and does not implement for incompressible ones; the -mentioned equation ensures the validity of the flow domain point.
Now that is all from the derivation using the divergence theorem. The next topic is also a part of it. Have a look.
The continuity theorem is defined in different ways, and one of them will describe by me. The continuity theorem starts with the control volume, and, taking it as a base, I will tell the whole topic.
So following are some essential key points related to the topic:
We will start with the assumption. Let us consider an infinitesimal box that controls volume and aligns with the axes in the Cartesian Coordinates system.
The following is the diagram that shows the box-shaped control volume.
As you can see from the diagram along the x-direction, the length is mentioned as dx, in the y-direction as dy and along the z-direction as the dz, respectively.
Moreover, at the centre of the box, there is information that says that the density is defined as a symbol and the velocity is defined in velocity components as u, v, and w, respectively.
So, we use Taylor's Theorem within the box at different locations away from the centre. And to define this point, we will use an example as follows:
(u)centre of right face=ρu+∂(ρu)∂xdx2+12!2(ρu)x2(dx2)2
In the case of the above-mentioned equation, the control volume has just limited to a point only, and the higher power or even the second power terms are negligible.
So, for the six faces of the box, we use the Taylor series expansion to density times the normal velocity component at the central point of each of the six faces, so they are as follows:
Center of Right Face
(u)centre of right face≅ ρu+∂(ρu)∂xdx2
Centre of left face
(u)centre of left face≅ ρu-∂(ρu)∂xdx2
Centre of Front Face
(w)centre of front face≅ ρw+∂(ρw)∂zdz2
Center of Rear Face
(w)centre of rear face≅ ρw-∂(ρw)∂zdz2
Center of Top face
(v)centre of top face≅ ρv+∂(ρv)∂ydy2
Center of Bottom face
(v)centre of bottom face≅ ρv-∂(ρv)∂ydy2
Now, there is another statement that says that
The mass flow rate into or out of the faces of the box is equal to the density times the normal velocity components at the centre point of the face times the surface area of the face.
We can define it as mathematically as follows:
m=VnA
The above equation is valid for each face. The Vn shows the magnitude of the normal velocity through the face, and the A shows the surface areas of the face.
The following diagram shows the mass flow rate through each face of our infinitesimal control volume; you will easily get the idea about it:
So all the theoretical background that I have mentioned in the diagram that is above can be easily understandable by the diagram.
For all the nonnormal velocity components, the truncated Taylor series expansions at the centre of each face can also be defined. But we have not represented as these components are tangential to the face.
Now we will move towards another equation, and that equation says the control volume shrinks at any of a point, and the value of the volume integral on the left side of the equation (b) will become as follows:
∫cv∂ρ∂t dV≅∂ρ∂tdxdydz
As we know that the volume of the box is dx, dy, and dz, respectively.
Here is a trick: with the help of the diagram, we can apply the approximations from the figure to the right side of the equation mentioned above. In this case, we will add all the mass flow rates at the inlet and the outlet of the control volume through all of the faces. Then take a left, bottom, and back faces contributing to the mass flow rate. Then concerning the equation, the right side will be as follows:
inm≅(ρu-ρu∂xdx2)dydz+ρv-ρv∂ydy2dxdz+(ρw-ρw∂zdz2)dxdy
The following are the faces that are mentioned in the equation:
Left Face =ρu-ρu∂xdx2dydz
Bottom Face=ρv-ρv∂ydy2
Rear Face=(ρw-ρw∂zdz2)dxdy
The next topic is also part of the Continuity Equation. So without wasting any time, let us start.
There is also an alternative way to present the continuity equation as we know that the following equation is according to the product rule of divergence theorem:
∂ρ∂t+V. V=∂ρ∂t+V. ρ+ρ.V=0
In order to explain the continuity equation alternative way, I will explain it in a few important key points:
The following is the alternative form of presenting the continuity equation:
1DρDt+.V=0
The above-mentioned equations show the fluid element that is flowing through the flow field, and it is also called the material element. Here there is a change that the .V is the change in the density.
So we can say that if the change in the density of the material element (fluid element) is small as compared to the magnitudes of the velocity gradient in the .V also if the element moves around, then the flow is said to be incompressible, and it is limited to it only.
So that is all from the alternative way of presenting the continuity equation. Now the next topic is another part of presenting the continuity equation.
The cylindrical polar coordinates system is a way of presenting the terms in (r,,z). it is also known as the cylindrical coordinates system.
In this topic, I will show you how the terms can be explained in a cylindrical system, and it is another way of presenting the continuity equation. So following are some of the important key points related to the topic. Have a look at it:
There can be three-dimensional, one-dimensional and two-dimensional cylindrical coordinates, but here in this topic at the start, I will explain in terms of the two-coordinates system only.
The following are the explanation of the (r, θ):
Here, r shows the radial distance i.e., from the origin to any point (let us say P).
Then there comes , now which shows the angular measurement from the x-axis (if we talk about generally, then is defined as a positive value, and it is in a counterclockwise direction).
The following is a figure that shows the complete explanation of the theoretical background:
There are more important terminologies that need to be defined. So following is the explanation of it:
ur and u are the velocity components
er and e are the unit vectors.
In the case of three-dimensional, there is obviously a z-axis. So in order to show the points, the following diagrams explain well. Have a look at it.
So as you can see that in the three-dimension case, there are two more values, one for the velocity component and the second for the unit vector.
So the following equation shows the coordinate transformation:
r=x2+y2
x=rcos
y=sin
θ=yx
So following is the equation that expresses the continuity equation in terms of cylindrical coordinates.
∂ρ∂t+1r(rρur)r+1r(rρu)+1r(rρuz)z =0
Thank you for reading.
It is one of the most critical topics whenever. It is related to the resistance a fluid faces in motion. A fluid exerts a force on a body in a different direction. Now the main question is, what is drag? And what do we know about it?
The definition of drag is as follows:
The force exerted on a flowing fluid in the direction of fluid flow is called drag.
Some of the essential key points related to the drag are as follows:
In order to elaborate on the drag force through an example. The body is attached to calibrated spring, and it is used to measure the displacement in the direction of flow.
The drag balances are one of those devices that are commonly used to measure the drag force.
It is not wrong to say that the drag force is much similar to the frictional force.
The more reduction in the drag force, the less fuel will be consumed.
An interesting fact about drag is that we can also produce a beneficial effect by drag, so we need to maximize its value. In this scenario, drag helps in pollen flying, usage of parachutes, movement of leaves and much more. These are one of the common examples used in our daily life.
The drag force is a combination of pressure and the wall shear forces in the flow direction.
In order to explain the lift in easy words, let us have a look at the definition.
The component of pressure and the wall shear forces in the normal direction to the fluid flow that tends to move the body in that direction is called lift.
Some of the important key points related to the lift is as follows:
Whenever we are dealing with lift and drag, these both have different responsibilities. In the case of two-dimensional flow, the resultant of the shear and the pressure forces are divided into two important components. The flow in one direction only is the drag force, whereas the normal flow is the lift.
The lift and drag have the equations through we can find the accurate values theatrically and can compare them with the practical ones.
To calculate the differential drag force, we have a differential area dA on a surface PdA and w so the equation will be:
dFD=-P dAcos +wdAsin
The differential lift force is as follows:
dFL=-P dAsin -wdAcos
By integrating the above two equations, we can get the total drag and lift forces that are acting on a body:
Drag Force
FD=∫AdFD=∫A(-Pcos +wsin )dA
Lift Force
FL=∫AdFL=-∫A(Psin +wcos )dA
The above equations show the skin friction (wall shear) and pressure, which contribute to the drag and lift.
The lift and the drag are a strong function of angle attack.
The drag and lift forces depend on the upstream velocity, density, size, shape, and orientation of the body. It is better to work with the dimensionless numbers that are used for the representation of drag and lift characteristics of the body.
The drag and lift coefficient equations are as follows:
Drag Coefficient
CD=FD12V2A
Lift Coefficient
CL=FL12V2A
Here A is the ordinarily Frontal Area.
The topic is related to drag and lift. In order to define the friction and pressure forces, I will explain in few key points:
The net force exerted on a body by fluid in the flow direction drag due to the combined effect of wall shear and pressure forces.
The part of due directly to wall shear stress called skin friction drag.
It is the part that is due directly to the pressure P, called the pressure drag.
The friction and pressure drag coefficients are presented mathematically as follows:
Drag Coefficients Friction
CD, friction=FD, friction12V2A
Drag Coefficient Pressure
CD,pressure=FD,pressure12V2A
So these are the drag coefficients of pressure and friction.
If the values of these coefficients are available, then it becomes easier for us to find the value of the total drag coefficient and total drag force. The formulae of both of these are given as follows:
Total Drag Coefficient
CD=CD,friction+CD,pressure
Total Drag Force
FD=FD,friction+FD,pressure
The friction of drag is the main component of the wall shear force, and this force is in the direction of flow.
In the case of a flat surface, the value of friction drag is zero, and it is normal to the flow direction. The value is said to be the maximum for a flat surface parallel to the flow.
With the increase in the viscosity, there is an increase in the drag.
Another exciting fact about it is that Reynold’s Number is inversely proportional to the viscosity of a fluid. So when the value of Reynold’s number is high, then the value of the total drag or the friction drag is less.
When the value of Reynold’s number is less, it is due to the friction drag and mostly happens in the streamlined bodies.
the bodies having a large surface area have a significant friction drag value. But it is independent of the surface roughness.
In the case of pressure drag, it is proportional to the frontal area.
The value of drag pressure for the blunt bodies is maximum; for the streamlined bodies is less, and in the case of the thin flat plates that are parallel to the flow, the value is zero.
It is also one of the important topics of drag.
The following are some important key points related to reducing drag by streamlining:
As I have discussed earlier that the drag pressure in the case of streamlined bodies is less.
Decrease the drag for a streamlined body by reducing the flow separation, ultimately reducing the pressure drag.
The streamlining delay the boundary layer separation resulting in a decrease in the pressure drag and an increase in friction drag.
The following diagram shows the difference in the values of friction, pressure, and total drag coefficients of a streamlined strut.
As you can see from the diagram, the value of total drag at minimum is D/L=0.25.
The drag coefficient value will be five times in the case of a circular cylinder having the same thickness as the streamlined shape.
In the case of an elliptical cylinder shape, the value of the drag coefficient is less. The elliptical cylinder shape is considered the perfect example for defining the effect of streamlining on a drag coefficient.
I will define flow separation extensively, so the following is the definition of it.
The fluid has high velocity when force flows over a curved body. Similarly, fluid can climb uphill on a curved surface without distraction.
At high velocities, the fluid stream detaches itself from the body’s surface, known as flow separation.
A flow can be separated from a surface when fully submerged in an immersed gas or liquid.
There are many essential terminologies for this topic, and the following are some important key points, equations, and definitions for the parallel flow over the flat surface. So let us start.
The following diagram shows a flat plate on which a fluid has flowed.
Taking this diagram as a reference, I will explain the whole topic through this.
Here, in this flat plate, the x-axis is measured along the plate surface (starting from the leading edge of the plate in the direction the fluid is flowing), whereas the y-coordinate is measured from the surface in the normal direction.
The surface’s velocity is equal to the velocity of the fluid that travels along the x-coordinate.
As you can see through the diagram, for our convenience, we have assumed that the fluid is in adjacent layers and they are piled onto one another
By doing so, the velocity of the first layer of fluid adjacent to the plate becomes zero, and this is due to the no-slip condition.
The first layer impacts the other layers by slowing the motion of particles of other different. And the process goes on as the layer slows down the next layer’s molecules.
The presence of the layer is then felt up to some normal distance (), from the plate, beyond which the free-stream velocity remains unchanged.
There are some critical regions and layers present on this layer, and they are defined as follows:
Velocity Boundary Layer
It is the region of the flow that is above the plate bounded by normal distance () in which the effect of the viscous shearing force caused by fluid viscosity is felt.
Irrotational Flow Region
It is the region where the frictional effect is negligible, and the velocity is constant.
The thickness of the boundary layer (δ) is the distance y from the surface at which u=0.99V.
There is a hypothetical line that is present on the layer (u=0.99V) that divides the flow into two regions, and they are named as follows:
Boundary Layer Region
Irrotational Flow Region
In this flat plate parallel to the flow condition, the pressure drag is zero, and the drag coefficient equals the friction drag coefficient. Mathematically we can present it as follows:
CD=CD,friction=Cf
Here Cf is the drag friction coefficient.
The equation for calculating the friction force on the plate is as follows:
FD=Ff=12CfAρV2
Here the A is the surface area of the plate.
From the diagram, you have observed that the velocity profile is in laminar and turbulent flows.
Also, the turbulent is much fuller than the laminar one. As it has four regions, and they are named as follows:
Viscous Sublayer
Buffer Layer
Overlap Layer
Turbulent Layer
The transition of laminar to turbulent flow is dependent on the geometry of the surface, roughness, upstream velocity, surface temperature, and many things.
The Reynold’s number at a distance x from the leading edge of a flat plate is as follows:
Rex=ρVx=Vxv
In the case of a flat smooth plate, the transition from a laminar to a turbulent flow starts at Reynold’s number RE≅1×105. The flow does not become turbulent until the value of Reynold’s number reaches RE≅3×106
Friction Coefficient
The following are some key points related to the friction coefficients:
In the case of laminar flow, we can calculate the value of friction coefficients by using the law of conservation of mass and momentum.
In the case of turbulent flow, it should be calculated experimentally and should be expressed in the empirical correlation.
The drag force for the whole surface can be calculated by using the average friction coefficient value.
In some cases, if we want drag force for a specific location, then in this condition, we must know the local value of the friction coefficient.
If we have the value of local values, then it becomes easy for us to calculate the average friction coefficient values:
Cf=1L0LCf,xdx
It is one of the important related to lift and drag. So following are the important key points related to the flow over the cylinder and spheres:
In our daily life, if we look around, there are multiple examples of it. In tubes (shell and tube heat exchanger) involves the internal and external flow over the tubes.
In sports, cricket, soccer, and tennis balls are the best examples of this topic.
For calculating the circular cylinder or sphere, the external diameter is taken as D.
In Reynold’s number, the formula is as follows:
Re=VDv
Here the V stands for the uniform velocity of the fluid as it approaches the cylinder and sphere.
Here, the value of the critical Reynold’s number is Recr≅2×105.
The change in total drag coefficient value is observed for the flow of cylinders and spheres.
It is not wrong to say that the drag force is due to the friction drag at the low value of Reynold’s number (Re< 10) and in the case of pressure drag at a higher value of Reynold’s number (Re>5000).
The following diagram shows the separation of the laminar boundary layer with the turbulent over a cylinder.
Effect of Surface Roughness
In the previous topic, I discussed this thing that the impact of having a surface makes a huge difference. So while discussing the cylinders and the spheres, it becomes important to discuss them in detail.
The following are some essential key points related to the topic:
The increase in surface roughness increases the drag coefficient in the case of turbulent flow.
For the streamlined case, it is also the same. But for the spheres and the cylinders, the increase in the roughness of the surface decreases the coefficient of drag. This means they have an indirect relationship.
The following diagram shows the indirect relation for cylinders and spheres.
The indirect relation is done by tripping the boundary layer into turbulence at the lower value of Reynold’s number. The result is that the fluid is close in behind the body, reducing the pressure drag force.
The value of Reynold’s number is Re≅2×105, and the value of the drag coefficient is CD≅0.1 in the case of a rough surface along with D=0.0015. In the case of a smooth surface, the values changes, and they become CD≅0.5.
The value of Reynold’s number for the rough sphere is Re≅106, and the drag coefficient value is CD≅0.4.
So the value of the drag coefficient for a smooth sphere is CD≅0.1.
The rougher the sphere will become, the more drag will also increase.
In order to exemplify the values, let us take an example of a golf ball. The velocity value ranges from 15 to 150m/s for the golf ball, and the value of Reynold’s number is 4×105,.
Lift
At the start of the article, I have already discussed what lift is. Here in this topic, I will explain extensively about the lift and the mathematical equation.
In order to explain the topic in a symmetric manner, I will explain it in key points, and these are as follows:
As you know that lift is the component of the net force, and this net force is because of the viscous and pressure force.
The coefficient of the lift is explained as follows:
CL=FL12V2A
The A here presents the planform area, and this area is viewed by someone that is looking at it from above in a direction normal to the body.
The V is the upstream velocity.
We will consider the airfoil with a width b and chord length c and the planform area as A=bc, respectively.
The following is the diagram that shows the airfoil structure.
Here, there is a term called wingspan or span. It is the distance between the two ends of the foil.
In the case of aircraft, the wingspan is the total distance between the tips of two wings.
Another important term is wing loading. It is the average lift per unit planform area FLA.
The airplanes are all based on the lift.
The purpose of discussing the lift in detail is to know how airfoils are designed and how they generate lift by keeping the value of drag minimum.
The streamlined bodies, such as airfoils that intends to generate the lift, have a negligible lift, and the wall shear is parallel to the surface.
I will show you the pictorial representation of the irrotational and actual flow past the non-symmetrical two-dimensional airfoils.
The following diagram shows the irrotational flow past a symmetrical airfoil (here the lift is zero).
The following diagram shows irrotational flow past a non-symmetrical airfoil (having zero lift).
The following diagram shows the actual flow past a non-symmetrical airfoil, and here the lift is positive.
Hope you enjoy reading the article. I have tried my best to explain you every point in easy words. Thank you for reading.
Hello Friends. I hope you are doing great. Here I am with another exciting topic of fluid mechanics, i.e. flow over bodies. In this article, I will explain the flow of fluids, the flow rate, their nature, the flow in pipes and much more. Fluid mechanics is all related to the fluid and its nature. I will discuss the forces that are on the body that is immersed in a fluid and the flow that is over the body. As the title shows, the main emphasis will be on the lift and drag forces. Moreover, the external and the internal flow will also be discussed in this article. And I am sure that you will enjoy reading this article. So without wasting any time, let us start.
I will start with an introduction to fluid flows.
In this article, our focus will be the fluid flow over the bodies immersed in the fluid. There are six different types of flows, and they are named as follows:
Steady and Unsteady Flow
Uniform and Non-Uniform Flow
One, two and three-dimensional Flow
Rotational or Irrotational Flow
Laminar and Turbulent Flow
Compressible and Incompressible Flow
Before discussing them extensively, I will briefly explain the external flow.
Let me explain the external flow in critical points:
We will take two situations. The first situation says that the body is at rest, and the fluid flow moves over it, and the second one says that the body is moving through some quiescent fluid.
If you look around, there are building standing still, and the wind is blowing over the building. This is the case when the body is stationary
The second condition is that the car is moving through the air.
So now, these are two different scenarios, but the main thing that matters is the relative motion of the fluid and the body. So, such motions are analysed by fixing the coordinate system on the body, called External Flow or Flow over the body.
Now moving towards the six types of flows, let us start explaining them one by one.
The definition of the steady flow is as follows:
The steady flow is where pressure, velocity, and cross-section differ from one point to another, and it does not change with time.
Some of the essential key point about the steady flow is as follows:
There are changes in the flow, but the change is so small that one particular parameter remains constant for a fixed period. But we can say that it is an ideal case because, in the actual case, there are very few chances that the parameters can remain constant.
The definition of the unsteady flow is as follows:
Changes in parameters at some stage can make the floe unstable. In practical cases, there is always a change in values of pressure and velocity values. Average values are constant; then, the flow is constant.
Some of the essential key point about the unsteady flow is as follows:
The unstable flow is said to be uniform and non-uniform both.
In the case of uniform flow, the cross-sectional area of fluid flow through the stream is said to be constant.
The definition of the uniform flow is as follows:
The distance along the flow path is constant then the flow parameter is also said to be constant. The fluid flow is called uniform flow.
Some of the essential key points about the uniform flow are as follows:
The cross-sectional area in the case of uniform flow is constant.
One of the best examples of uniform flow is the flow through the pipeline.
The definition of the non-uniform flow is as follows:
The non-uniform fluid flow is due to the variations in the flow parameters, varying at different points on the flow path.
Some of the essential key points about the non-uniform flow are as follows:
It is evident, as the names show, that the parameters do not remain constant. A change in velocity can be observed; ultimately, the flow is also said to be non-uniform.
In practical cases, the flow near the solid boundary is said to be a non-uniform flow.
The term has three types of fluid flows.
Starting the one-dimensional flow its definition is as follows:
The flow that travels in one dimension and changes parameters such as pressure, velocity, depth etc., in one flow direction only is said to be one-dimensional flow.
Some of the essential key points about the one-dimensional flow are as follows:
The change in parameters in one direction at a given moment is considered a one-dimensional flow.
There are many chances that the flow can be unstable if the parameter alters in time, but it is still not across the cross-section.
The definition of the two-dimensional flow is as follows:
The flow parameters differ in the flow direction, and in one direction in the right angles, the flow is then said to be two-dimensional flow.
Some of the essential key points about the two-dimensional flow are as follows:
Toto elaborates on the two-dimensional flow; following diagram explains the flow well. Have a look at it.
The flow is either in the x direction or y. there is no change in the z-direction.
In two-dimensional flow, streamlines are usually curved in one plane and the same on every parallel plane.
The definition of the three dimensional flow:
The element of the fluid moves in three dimensions (translation, rate of deformation and rotation) in space. Such a kind is called the three-dimensional flow.
Some of the important key points related to the three-dimensional flow is as follows:
The following is the diagram that shows the three-dimensional flow:
Let us start with the rotational flow:
The definition of the rotational flow is as follows:
The flow in which the fluid particles rotate on their axis while along the flow line is called the rotational fluid flow.
The definition of fluid flow is as follows:
The flow in which fluid particles do not rotate in their own axis when theyw along the flow lines.
The visual representation of both flows is as follows;
The following is the diagram that shows the rotational fluid flow:
The following diagram shows the irrotational fluid flow:
Let us start with laminar flow
The definition of the laminar flow is as follows:
The flow in which fluid particles move in a well define path and the streamlines straight and parallel. Such kind of flow is called laminar flow.
Some of the important key point related to the laminar flow is as follows:
The pictorial representation of the laminar flow is as follows:
The definition of the turbulent flow is as follows:
The flow in which the fluid particles do not move in a well define path and it lead to high energy losses throughout.
Some of the important key point related to the turbulent flow is as follows:
The pictorial representation of the turbulent flow is as follows:
The definition of the compressible fluid flow is as follows:
The flow in which the density of fluid varies from one point to another point (not constant) is called the compressible flow.
The pictorial representation of the compressible flow is as follows:
The definition of incompressible flow is as follows:
The flow in which the density of the fluid flow is constant is said to be incompressible flow.
The pictorial representation of the incompressible flow is as follows:
That is all from the types of fluid flow. Now, we will move towards the main topic, i.e. lift and drag. Let us start.
Hello Friends. I hope you are doing great. In this article, I will cover the nature of flow responsible for the changes in density. The nature of flows responsible for density change is known as compressible flows.
Fluid dynamics and thermodynamics combine for the theory of compressible flows. The compressible flow creates a relationship with the ideal gas. First, I will explain the stagnation state, speed of sound, and then the Mach number. This all comes under compressible flows. Then there come the static and stagnation fluid properties, shock waves, and variation of fluid properties in the case of regular and oblique waves. All of these are discussed in this article. So without wasting any time, let us start with the Stagnation property.
The definition of the stagnation property is as follows:
The enthalpy and kinetic energy of the fluid is combined in a term called stagnation or total enthalpy.
The formula of Stagnation Property:
The procedure of stagnation property is as follows:
ho=h+V22
Units of Stagnation property:
The units of stagnation property are kilo Joule per kilogram.
Some of the essential key points related to the stagnation property are as follows:
In the definition of stagnation property, there is a term enthalpy. The enthalpy is the combined flow and internal energy of the fluid. The formula of enthalpy is given below:
h=u+P
It is better to say that enthalpy is equal to the total energy in the case when the potential and the kinetic energy of the fluid are not considered (or negligible).
So, when there are high fluid flows (usually in jet flows/high flows), the potential energy is not considered, whereas kinetic energy is thought. In this situation, the enthalpy and the kinetic energy combine and are called the stagnation enthalpy.
Whenever the fluid's potential energy is not considered, the total energy is said to be stagnation enthalpy.
To differentiate the stagnation enthalpy from the standard enthalpy, the ordinary enthalpy is known as static enthalpy. In contrast, both enthalpies become the same when the kinetic energy is not considered.
Derivation of Stagnation Enthalpy
I will share some essential points related to stagnation enthalpy in this part. So without wasting any time, let us start.
In order to start the derivation of the stagnation enthalpy, we will consider the preferably steady flow, which flows from a nozzle or diffuser. The main property of the flow is that it is flowing on its own, not with the aid of a motor or anything.
The same thing is also implemented; the potential energy is considered minor or negligible.
In this case of a single stream of the steady flow device, the energy balance equation is said to be as follows:
h1+V122=h2+V222 …(a)
h01=h02 (b)
It is better to say that the above equation represents that during the single stream steady flow, there is no change of potential energy, heat, and work interaction is also not present.
A decrease in static enthalpy is observed when there is an increase in velocity. This usually happens in devices that are diffusers and nozzles, respectively.
Another equation is observed, and from equation (a), the velocity of state two will equal zero when the fluid is stopped. So the equation transforms into new, and that is as below:
h1+V122=h2
h02=h1+V122
Another exciting fact says that when there is no heat transfer from the system to the system, it is called an adiabatic state. So the stagnation enthalpy is also considered to be the enthalpy of fluid.
Pressure and temperature also increase when there is the conversion of kinetic energy to enthalpy.
You might be wondering from the heading that says Stagnation Properties. So Stagnation Property is the property of fluid at the stagnation state.
You have observed from equation (b) that there is a transcript 0. So actually, 0 represents the stagnation stage.
Here is another term called isentropic stagnation state and the definition of it is as follows:
The isentropic stagnation state is the one when the stagnation process is adiabatic as well as reversible.
one of the salient features of the isentropic stagnation state is that the entropy remains constant, so in the case of an actual stagnation state, the entropy might increase or decrease depending on the situation.
Both the stagnation state and the isentropic stagnation state are considered the same sometimes.
There is another equation that says that in the case of an ideal gas, equation (b) will become as follows:
CpT0=CpT+V22
The final equation after dividing Cp from both sides is:
T0=T+V22Cp
Here T0 is said to be the stagnation temperature ( transcript with 0 presents stagnation state).
T0 is the temperature at which ideal gas adiabatically becomes a state of rest.
The next term is stagnation pressure. Before defining it, let us see the equation first, and it is as follows:
P0P=(T0T)k/(k-1) (c)
So stagnation pressure is defined as the pressure of a fluid that becomes in a rest state adiabatically.
equation (c) presents the stagnation pressure.
At the start of these critical points, I have used a term, i.e. single stream steady fluid flow. So for this term, the energy balance equation becomes:
qin+win+h01+gz1=qout+wout+(h02+gz2)
The equation in the case of an ideal gas with the specific heats will be as follows:
qin-qout+win-wout=CpT02-T01+g(z2-z1)
So here h02 and h01 are the stagnation enthalpies, whereas T02 and T01 are the stagnation temperature
So this is all from the stagnation enthalpy, property, temperature, and property.
Now I will explain what Mach number is and the speed of sound. So let us start.
Both are interrelated to, first, the speed of sound.
Speed of Sound
After reading the heading, you might be wondering why the article is about compressible flow low then why we are discussing the speed of sound. So better not to worry and let me explain to you in easy words.
The speed of sound in terms of compressible flows is explained as follows:
The sound rate is an integral part of the study of compressible flows as it is the sonic speed at which infinitesimally small pressure waves travel through a medium.
Some of the essential key points and the equation are discussed below. Have a look.
If you look into the definition, there is a statement that says that small pressure waves travel through some medium. Here the pressure waves are created from turbulence or some disturbance.
To explain the topic more precisely, I will explain the speed of sound through some theoretical derivation background.
There is a duct has fluid in it, and that fluid is at rest. There is also a piston inside the duct, which is moved inside. The piston is being moved at some velocity known as constant incremental velocity.
This disturbance of the piston moves the waves of fluid and the movement is made at the speed of sound.
The waves are separated from the fluid adjacent to the piston thus resulting in a change in thermodynamics property.
To elaborate on the whole scenario, the following diagram is presented below:
For a more precise explanation a control volume is considered., In this situation, the waves created by the disturbance or any movement inside remain within the boundary.
Here mass balance equation for the single stream, steady flow is as follows:
mright=mleft
And on simplification, the equation will become as follows:
ρAc=ρ+dρA(c-dV)
Let me explain this equation to you. As you know that in the above theory we have considered a control volume. So when something passes through, there are some movements of the waves, and the fluid moving in the right direction will now move in the direction of the wave front. The speed at which the fluid moves is the speed of sound c. Now the fluid moving in the direction of the left will move away from the wave front, and now rate will be c-Dv, respectively.
So I hope that now the equation is quite understandable to you.
we can further simplify the equation by deduction of cross-sectional area, and now the equation will become:
cdρ-ρdV=0 (a)
The definition of control volume says that there is no heat transfer and work done throughout the process. The equation will not have the potential energy for the steady flow process, and it will be as follows:
h+c22=h+dh+(c-dV)22 (1)
And after the simplification, the equation becomes as follows:
dh-cdV=0
You might be shocked at how the equation is reduced so much as it has so many terms before. So here is the answer. In equation (1), the term dV2 is not considered.
The ordinary sonic wave is nearly unneglectable. They are small, yet they are still considered. So they are somehow isentropic. Now the equation in the case of thermodynamics will be as follows:
Tds=dh-dP (b)
Here the term on the left Tds will be equal to zero, and after the simplification, the equation will become:
dh=dP (c)
For the calculation of the speed of sound, we will combine the equation (a), (b), and (c) will become as follows:
c2=dPdρ
After the simplification, the final equation for the speed of sound will be as follows:
c2=k(∂P∂ρ)T
So here is the final equation of the speed of the sound.
We are moving towards the second part, the Mach Number. Let us see the definition and the equations of it.
Mach Number
The Mach number is also one of the essential parts of compressible flows.
The definition of the Mach Number is as follows:
The Mach Number is the ratio of the actual velocity of the fluid to the speed of sound in the same fluid.
Formula of Mach Number:
The formula of the Mach number is as follows:
Ma=Vc
Some of the essential key points of the Mach number are as follows:
The Mach number has a direct relationship with the velocity of fluid means an increase in the Mach number increases the velocity and vice versa. Whereas the Mach number has an indirect relationship with the speed of sound. The increase in one quantity decreases the other.
There are some important parameters for the flow regimes, and they are as follows:
In the case of
Sonic Mac=1
Subsonic … Mac<1
Supersonic Mac>1
Hypersonic Mac≫1
Transonic Mac≅1
While discussing the flow type, the one-dimensional isentropic flow comes under the compressed fluids.
Examples:
The nozzle diffusers and the turbines are examples of one-dimensional fluid flow.
Some of the essential key points related to the one-dimensional isentropic flow are as follows:
I will explain to you the theoretical background of the one-dimensional flow. Before that, look at the diagram as it will be convenient for you to understand the theory.
As you have seen the diagram and know about the number, let me explain. The value of the Mach number is one at the smallest flow area, called the throat.
Increase in fluid velocity is observed when the fluid passes through the throat. But this increase in velocity is also due to the decrease in the density of the fluid.
If you have a look at the diagram, then there is a word i.e. converging nozzle. But here the converging-diverging nozzle is used so it is the one in which the flow area first increases and then decreases at the throat. Some might confuse these converging nozzles with the Venturi ones. But the Venturi nozzles are used for the incompressible flow.
In this topic, we will pinpoint the main parameter that involves the coupling in the case of isentropic duct flow between the density and velocity.
In this area, we will look forward to some equations of steady flow and the changes in the fluid velocity accordingly to the flow area. So following are some essential key points related to it:
The equation will present the relation between the temperature, pressure, density velocity, Mach number and flow area. These are the equation for the isentropic one-dimensional flow.
The following is the mass balance equation in the case of a steady flow process:
m=ρAV=constant (a)
The equation (a) will be differentiated and divided and the following result will be obtained;
dρ+dAA+dVV=0 (b)
Now the next equation will be related to the isentropic flow in which there is no potential energy and it is as follows:
dP+VdV (c)
The next equation is expressed in terms of Bernoulli’s Equation and again the potential energy is neglected. For convenience, we will combine the equation (b) and (c) and it will become:
dAA=dP(1V2-dρdP)
Now here comes the final equation that describes the isentropic flow in the terms of changes in the pressure accordingly to the flow area.
dAA=dPV2(1-Ma2)
The above-mentioned equation has everything it has an area, density, pressure and Mach number
For this equation there are some limitations in the case of Mach number and they are mentioned below:
Subsonic Flow … Ma<1 and (1-Ma2)term is Positive
Supersonic Flow Ma>1 and (1-Ma2) term is negative
Let me explain these limitations more extensively. In the case of subsonic, the value of differential is dA and differential pressure dP should have the same signs. The increase in pressure increases the flow area and vice versa. Meanwhile, in the case of a supersonic, the value of a differential area and the pressure should have opposite signs. Also, an, increase in pressure quantity should have decreased the flow area of the duct.
There is another equation that is also related to the isentropic flow and it says in equation (c) the value of density and value is substituted in the final equation and it is as follows:
dAA=dVV1-Ma2
So ending this topic on the limitations of Mach number for this equation and they are as follows:
Subsonic Flow (Ma<1) dAdV<0
Supersonic Flow (Ma>1) dAdV>0
Sonic Flow (Ma=1) dAdV=0
Isentropic Flows for Ideal Gas
While discussing the isentropic flow this topic holds its importance as related to the ideal gas. Let’s have a look at the equations and some important key points of it:
For ideal gases both the static and the stagnation properties are combined together, and with the help of the Mach number and specific heat, the equations will be transformed.
The first equation is about the stagnation temperature T0and the ideal gas temperature T. And the equation will be like this:
T0=T+V22Cp
And after simplification, the equation will become:
T0T=1+V22CpT
In this equation, we will substitute the values of V and specific heat. The values we will be substituting are:
Cp=kRk-1
c2=kRT
Ma=Vc
After substation of all the values and doing the simplification, the final equation will become:
T0T=1+(k-12) Ma2
If you remember that at the start, I used two terms, i.e., static and stagnation pressure. So now, I will show you one by one equation of both these terms, and they come under ideal gas. So presenting you the ratio of stagnation to the static pressure, and the equation is:
P0P=[1+k-12Ma2]kk-1
This equation is the result of substituting the value of T0T in this equation.
P0P=(T0T)k/(k-1)
Now presenting you the ratio of stagnation to static density:
0=[1+k-12Ma2]kk-1
There is some limitation where the value of the Mach number is one, so all the points where the Mach number is unity are called the Critical Properties.
While writing the equations of compressible flows the asterisk( *) is used to present the critical points. So their equations will be as follows:
T*T0=2k+1
P*P0=(2k+1)k/(k-1)
*0=(2k+1)1/(k-1)
So these are the equations that present the critical points. That is all from the one-dimensional isentropic flow.
In compressible flows, isentropic is considered to be one of the most important branches of it. Here under this topic, I will discuss the isentropic flow that passes through the nozzles, what are their nature equations, and the nozzle type. Let us start.
First of all, I will share some of the essential key points with you that covers all the information about the topic.
Here in this topic we will learn a new term i.e. Back pressure. A brief definition of back pressure is as follows:
Back pressure is pressure that is applied at the discharge region of the nozzle.
Here the nozzle can be converging and diverge depending on its requirement of it. I will discuss the nozzle type and the isentropic flow through it.
Converging Nozzle
The derivation and the equation of the converging nozzle are as follows:
A subsonic flow is considered in the case of a converging nozzle. The diagram elaborates on the flow:
Here you can clearly see that the inlet of the nozzle is connected to the reservoir. The temperature is presented as Tr whereas the pressure is as Pr respectively.
the velocity at the inlet is neglected and the flow is isentropic. The reservoir has a huge volume so that is the main reason the fluid velocity’s not being considered.
The reservoir pressure and temperature are equal to the nozzle’s pressure and temperature.
Now the working is being followed and the back pressure is being reduced. After doing so the change in the flow pressure is observed.
The back pressure is equal to the P1 and that P1 is equal to the reservoir pressure. This is all due to the uniform distribution of pressure among the nozzle.
Similarly when the back pressure is reduced to P2 then the value of at the outlet becomes equal to P2 also. As a result of this procedure, the pressure of the nozzle decreases in the direction of the flow.
This all working is elaborated by the diagram mentioned below;
The back pressure is now reduced at pressure point three and that is equal to the critical pressure. Actually, pressure three equal to the critical pressure shows that the pressure that is required by the fluid increases the velocity of the fluid and this increases the speed of sound at the throat.
Now the back pressure is reduced to pressure point four. Here there is no change in the pressure and the flow is being observed as it is at the end position.
Now we will look forward to the equation that is related to the mass flow rate.
m=ρAV
m=PRTAMakRT
m=PAMakRT
We will do further simplification and the final result will be as follows:
m=AMaP0k(RT0)[1+(k-1)Ma2/2](k+1)/[2k-1]
Whenever there is a Mach number there are certain limitations as discussed above according to the equations. Here when there is a specified flow area the mass flow rate can be determined by differentiating the above equation and equalizing it to zero. The value of the Mach number here is one. We all the when the value of the Mach number is one (it is at the throat) and the mass flow is maximum at the throat.
Now the equation will be as follows:
m=A*P0kRT0(2k+1)(k+1)/[2k-1]
In order to have a maximum flow rate there are two parameters to be fulfilled. The temperature and the pressure at the inlet of stagnation of a given throat give the maximum flow rate in the case of an ideal gas.
It is a characteristic property that by changing the stagnation pressure and temperature we can control the flow rate values.
So these are the theoretical background and equation related to flow rate. Hence there is another point related to this topic and that is the critical Mach number.
The equation for it is the same as the formula for the Mach number but there are a few modifications, and they are as follows:
Ma*=Vc*
The formula can be expressed in the term of temperature, and the working is as follows:
Ma*=Vc.cc*
Ma*=Ma. cc*
Ma*=MakRTRkT*
Ma*=MaTT*
We can further simplify this equation and can get the final critical Mach number equation so by substituting the values of local and critical temperature, we get:
Ma*=Mak+12+(k-1)Ma2
So, this is the final equation. There is a difference in both the values of the Mach number.
The critical Mach number Ma* is the local velocity non-dimensionalized concerning the sonic velocity at the throat. The other one Ma is the local velocity non-dimensionalized concerning the local sonic velocity.
Their values also differ from one another. The value of Ma* is provided on a table that is listed against the Ma when the value of k is 1.4.
The table is shown below for your clarification:
So that is all from the isentropic flow through nozzles. Let’s move toward the next topic, related to the waves.
Where ever there is flow, there are waves created. As we have studied in depth how sound waves are made, the question is how shock waves are formed. Let us have a concept about it.
There are changes in the fluid properties that can either be temperature pressure, increase or decrease in volume, etc. These changes occur in a thin layer of the converging-diverging nozzle under supersonic flow conditions. So in this way, shock waves are created. Now I will explain the type of shock waves, their working principle, and the equations. So without wasting any time, have a look at it.
Normal Shock Wave
The definition of normal shock waves is as follows:
The normal shock waves are the ones that form in a plane normal to the direction of flow.
Some of the essential key points, along with the equations of normal shock waves, are explained below:
The shock wave is all irreversible and not being approximated as being isentropic.
The process includes all the energy, momentum, and conservation of mass relation to define the flow properties before and after shock.
The normal waves are different in the sense that they are thin and this property benefits the entrance and exit flow areas. The exit and the entrance flow area are the same.
Now moving towards the mathematical part. Let’s start with the conservation of mass. As you all know about the formula of conservation of mass but for convenience let me show you.
1AV1=2AV2
The equation can further be simplified with the cancellation of cross-sectional area.
For the conservation of energy, the equation is as follows:
h1+V122=h2+V222
Now let’s have an overview of the conservation of momentum:
AP1-P2=m(V2-V1)
While working on the equation for the normal shock waves we combine the law of conservation of mass and energy into one equation.
The equation is then plotted on the h-s diagram.
If you look at the diagram below, it shows all the important aspects that I am going to share.
The h-s diagram shows a line i.e., Fanno Line and that is the resultant curve. This curve has similar values of stagnation enthalpy and mass flux.
Moving towards the second curve i.e. the Rayleigh Line. There are two point on the h-s diagram and these two points are correspondence to the Mach (Ma=1).
Further you can see in the diagram that the upper part is for the subsonic flow whereas the lower part is for the supersonic flow.
The two lines Fanno and the Rayleigh line intersect at two points as you can see from the diagram. One point presents the condition that is before the shock and the second shows after the shock condition. At point 1, it is supersonic and before shock condition, whereas at point 2 it is subsonic and aftershock condition.
Also the high is the value of the Mach number before the shock there is a chance that the shock will occur.
When the Mach number is unity then the shock waves are simply sound waves.
One of the important points related to this is that the conservation of energy principle demands that the stagnation enthalpy should be constant throughout the process.
h01=h02
In the case of an ideal gas, the enthalpy is as follows:
T01=T02
After the shock, the temperature of the ideal gas remains constant.
Moving forward, there are some important equations that are related to the ideal gas. These equations show the before and after shock and after substituting the values required results can be obtained.
So following are two important equations.
T01T1=1+(k-12)Ma12
T02T2=1+(k-12)Ma22
We will further simplify the equation by taking T01=T02 and the result will be:
T2T1=1+Ma12(k-1)/21+Ma22(k-1)/2 (a)
In the case of an ideal gas, we know that:
1=P1RT1
2=P2RT2
The value of temperature can be easily substituted using the law of conservation of mass while the values of Mach number and c are used which we can substitute in equation(a). the values are as follows:
Ma= V/c
c=kRT
Now we can substitute all the values and get the result as follows:
T2T1=P2V2P1V1=P2Ma2c2P1Ma1c1=P2Ma2T2P1Ma1T1=(P2P1)2(Ma2Ma1)2 (b)
Now this is the final result we get after substituting all the values. We can further simplify this equation and can get the equation for the pressure ratio across the block and for that we will combine the equation (a) and (b):
P2P1=Ma11+Ma12(k-1)/2Ma21+Ma22(k-1)/2
If you recall the law of conservation of mass and energy, then after looking at the equation you will understand that it is the combination of laws.
Thus this equation about the Fanno Line for ideal gas.
The other curve is Rayleigh Line and its derivation is as follows:
P1-P2=mAV2-V1
P1-P2=2V22-1V12
As:
V2=(PRT)(Ma.c)2
V2=(PRT)(MakRT)2
V2=Pk Ma2
The pressure ratio can be as follows:
P2P1=1+kMa121+kMa22
Now we will move towards the final equation that is about the Fanno and the Rayleigh lines relating the Mach number upstream of a shock to that downstream of the shock. Let’s us have a look at an equation:
Ma22=Ma12+2/(k-1)2Ma12k/(k-1)
The oblique shock is a type of shock waves. It is better to say that there are various variations of waves. So moving toward the topic in order to define the oblique shock I will explain it through an example. So have a look at the example:
Example:
The space shuttle traveling through the atmosphere at supersonic speed creates a complicated pattern with shock waves. Such complex patterns and shock waves are called Oblique Shock Waves.
The following is the diagram that elaborates the shock waves:
Some of the essential key point and equations related to the oblique shock waves is as follows. Have a look at it:
Here in the diagram you can see there are some certain angles mentioned and they these angles helps in the formation of the oblique shock waves.
So talking about the first angle i.e. the theta and this is named as a turning angle or angle of deflection. The background of this angle is that at first, we consider the oblique shock waves traveling in a straight line. This straight line is much similar when the Ma>1 and in supersonic flow, a two-dimensional wedge half angle is formed. So here the fluid cannot flow through the wedge and at that point, it turns into the turning angle.
The other angle is about the shock angle or wave angle (). The background of this angle is that when the turning angle is formed then the straight oblique shock wave is formed and this wave is aligned with the wave angle.
For the conservation of mass, the wave angle should be greater than the two-dimensional half-wedge angle.
In the case of supersonic flows, Reynold’s number has a high value, and the boundary layer along the wedge is thin so its effects are not considered.
Now the flow is turned by another angle that is much similar to the wedge angle and it is named a deflection angle .
The deflection angle is the same as the half wedge angle.
Here there is an interesting fact about the half wedge and deflection angle i.e. when the thickness effect of the boundary layer is considered then the deflection angle is observed to be greater than the half wedge angle.
In the case of the oblique shock waves there are also the limitations of the Mach number. A decrease in the Mach number is observed in oblique shock waves.
Also oblique shock waves are possible when the upstream flow is supersonic.
Now moving towards some mathematical part of the oblique shock waves. The straight oblique shock wave is considered.
The diagram below shows the straight oblique shock waves:
Now here you can see that the straight wave is decamped into the velocity vector upstream and downstream of the normal and tangent components.
For working a small control volume is being considered and all the working happens on it.
If you look at the diagram closely, the upstream of the shock, all the fluid properties (such as pressure velocity) along the lower left face of the control volume are similar to those of the upper right face.
In this case, the law of conservation of mass equation is changed into the following equation:
1V1,nA=2V2,nA
Canceling the area of the control surface (A) on both sides we will get:
1V1,n=2V2,n
There is another important point related to the tangential component of velocity. So the tangential component of the velocity is parallel to the oblique shock and in order to prove this point the tangential momentum equation to the control volume is applied.
So conservation of momentum is being applied in the normal direction to oblique shock waves. So the equation we get is as follows that presented the forces that are only pressure forces:
P1A-P2A=ρV2,nAV2,n-ρV1,nAV1,n
P1-P2=2V2,n2-1V1,n2
Since we have considered a small control volume so we know that there is no transfer of heat and no transfer of work done. The stagnation enthalpy also does not change and eventually, the conservation of energy equation becomes as follows:
h01=h02=h0
h1+12V1,n2+12V1,t2
h2+12V2,n2+12V2,t2
Here the V1,t=V2,t. The equation will be reduced as follows:
h1+12V1,n2=h2+12V2,n2
Here is the final equation.
You might have observed that the equation of the conservation of mass, energy, and momentum have the same equations but when we are writing them in the terms of normal velocity components then they vary. But the equation for the normal shock is the same.
The equations that are derived for the normal shock waves can be applied oblique shock waves. Here is a condition that the oblique waves equation should be written in the terms of Mach number.
So we come to the conclusion that the equation for the oblique shock waves is much similar to the normal ones but the only difference is that the normal components of the Mach number are used in oblique ones.
It is also one of the important types of waves and it may differ from other ones also. Let us share some of the important key points related to it;
One of the important points about the supersonic flow is that when it travels in the opposite direction just in the upper portion of a two-dimensional wedge at an angled attack is greater than the half-angle sigma (δ).
The flow of such kind is known as expanding flow. But when the flow is in oblique shock then the flow is called the compressing flow.
The diagram shows the whole flow type and the angles created by it.
So there is not a abrupt change in the turning of flow by the shock but the Mach wave turn and thus the flow also turn by infinitesimal amount.
One of the important information about the flow is that the flow is isentropic.
In the downstream direction, a decrease in the pressure, temperature and density is observed whereas the Mach number increases.
Just have a look at the diagram you will see that Prandtl-Meyer is inclined at some angle called the Mach angle (μ) also the first expansion wave has an angle and we call it Mach angle. This angle is determined in the following way:
1=(1Ma1)
2=(1Ma2)
Now there is important to question how can we determine the Mach number 2. So for Ma2, there is a formula for it and that formula shows a turning angle across the expansion fan and can easily be calculated by integration. So the formula is as follows:
=v(Ma2)-v(Ma1)
Now, this is the formula for the turning angle across the expansion fan and can be used to find Ma2.
Here in this formula vMa is the Prandtl-Meyer function and it has an equation where we can find the unknown. So the equation is as follows:
vMa=k+1k-1( k-1k+1(Ma2-1))-( Ma2-1)
Here is an important point and i.e. that vMais an angle and it should not be confused with kinematic viscosity.
When the Mach number is unity then the Prandtl-Meyer function vMa taken as zero for getting the supersonic Mach number (Ma>1)
One of the interesting features of the Prandtl-Meyer function is that it has numerous advantages in our daily life.
The shock wave and boundary layer both have a unique significance. There are numerous advantages of both of these in our daily life.
The following are some important applications of shock waves and boundary layers. Have a look at them:
The boundary layer is susceptible to separation from aerodynamics whenever strong adverse pressure occurs
The shock wave on the other hand produces a strong pressure gradient.
Whenever the boundary layer and the shock wave interact with each other the result is a complex flow pattern and in some cases, the boundary layer separates from the region on which it is attached
The interaction between the shock wave and the boundary layer is viscous and inviscid interaction in which the viscous flow in the boundary layer encounters the essentially inviscid shock wave that is generated in the free stream.
The change in the shock wave and the boundary layer continues until an equilibrium condition is reached.
The shock wave and boundary layer have become one of the most pacing problems of modern fluid mechanics research.
Conclusion
Dear friends I hope you have enjoyed reading the article. I have tried best to explain extensively about the compressible flows, isentropic flows, shock waves, boundary layers, and the applications of both of them. That’s all my side. Thanks for reading.
The first law of thermodynamics is also known as the Conservation of energy Principle. I will explain what is energy principle is and the following are some important points of the Energy Principle:
The energy principle can be described as the following equation:
Ein-Eout=∆E
I will exemplify the energy equation through a simple example. For instance, the rock fell from the cliff and while falling from the cliff the rock gained speed and the speed doubled while coming down. The potential energy is converted into kinetic energy as the rock is falling from height. Here the air resistance is negligible here. All these factors confirm the conservation of energy principle.
One of the simple examples of the Conservation of Energy Principle in our daily life is the man who has a lot of energy i.e. the input. If he eats junk and gains weight and the energy will be stored in the form of fat also he exercises less and that is the output. But on the other, a man who eats less has less input and ultimately loses weight. So his input is small as compared to the output.
The conservation of energy principle states that
The change in energy content of a system is equal to the difference between the energy input and output.
If the mass, energy or other quantity enters the system from outside to the system and exists in the system. But if the quantity moves within the system then it is not considered to be a transferred quantity. So it is important to understand the boundaries of the system.
In a closed system, the energy can change to heat transfer and work transfer. In this case, the conservation of energy of the close system equation will be expressed as follows:
Qnet in+Wnet in=dEsysdt
The value of the net heat transfer rate is as follows:
Qnet in=Qin-Qout
The value of net power output is as follows:
Wnet in=W in-Wout
Here dEsysdt is the rate of change of the total energy content of the system.
In the case of a compressible system, the equation of total energy consists f internal kinetic and potential energies and the equation is as follows:
e=u+K.E+P.E
e=u+V22+gz
The energy transfer by heat is one of the important topics of the energy equation. The following are some of the important key points related to the energy transfer by heat:
The heat is in the form of a latent form of internal energy.
Heat transfer is defined as the movement of thermal energy by means of nature and the temperature is decreased in this case. So the transfer of thermal energy from one system to another due to temperature variations is known as the transfer of heat.
In the case of heat transfer, the heat is always transferred from a body of higher temperature to a body of lower temperature. When both temperatures maintain equal temperature then the process of transfer stops. In the case of bodies having equal temperature, there is no transferring of heat.
Adiabatic process is the one in which there is no transfer of heat. The system is said to be in adiabatic if the system is insulted and if the system and the surroundings are at the same temperature.
The adiabatic process is confused with the isothermal process. But there is no transfer of heat but in the adiabatic process, the temperature can be changed by the work transfer.
The energy transfer by work is also an important part of the energy equation. The following are some of the important key points related to energy transfer by work:
The energy interacts and results in the work.
The time rate of doing work is called the power and the power is denoted by the W.
The energy is increased when the work-consuming devices transfer energy to the fluid. A simple example of transferring energy is an electric fan. This moves the electric energy is converted into mechanical energy and this mechanical energy operates the motor of the fan and that motor starts the blades and the fan starts. But here we can say that energy is transferred and this energy transfer has no relation to the temperature.
The system has various kinds of transferring of work and the total work equation is as follows:
Wtotal=Wshaft+Wpressure+Wviscous+Wother
The shaft work is also one of the important part of energy equation. So following are some of the important key points related to the shaft work:
Machines such as turbines, pump, fans or compressors all have shaft and the work protrudes through the control surface by the shaft is known as shaft work.
The equation of the shaft work is as follows:
Wshaft=ωTshaft
Wshaft=2πnTshaft
Here omega is the angular speed of the shaft and n is the number of revolutions of the shaft per unit time.
The work done by the pressure forces is also one of the important topic that comes under the energy equation. So following are some important key points related to the work done by pressure forces:
In the case of work done by pressure forces, let us consider a gas that is compressed in a piston cylinder.
The diagram of gas compression in a piston cylinder is as follows:
The piston is moved at some distance and that differential distance is ds. The differential distance is under the effect of pressure forces PA where A is the cross sectional area of the piston whereas P is the pressure forces. Here the work is being done.
The equation for work done on the system is as follows:
Wpressure= δWboundry=PAVpiston
Here the value of Vpistonis as follows:
Vpiston=dsdt
We will start the derivation with a little theoretical background.
Let us consider a system that has arbitrary shape and the diagram is as follows:
The system moves with the effect of pressure. The pressure here acts in an inward direction and normal the surface. The equation for the time rate at which work is done by pressure on the differential part of the system is as follows:
Wpressure=-PdAVn
Wpressure=-PdA(V.n)
The equation for the normal component is as follows:
Vn=Vcos
Vn=V.n
So now I will show you the equation for the total rate of work done by the pressure forces:
Wpressure net in=-∫A P (V.n)=-∫APρ(V.n)dA
Now there is an equation that shows the general form of energy equation that is applied to the fixed, moving and deforming control volumes as follows:
Qnet in+Wshaft, net in+Wpressure, net in=ddt∫cvedV+∫cse(Vr.n)dA
So is the energy equation for the fixed, moving and deformed control volumes.
The statement of Bernoulli’s Equation is as follows:
In fluid dynamics, Bernoulli’s Equation states about an increase in speed of fluid that occur simultaneously with the decrease in static pressure or decrease in fluid’s potential energy.
After reading the statement, you might have little idea what Bernoulli’s equation is about. Don’t worry. Here I am to explain that to you extensively.
Some of the important critical points related to Bernoulli’s equation are as follows:
Bernoulli’s equation is the relation between the pressure, velocity and potential energy.
This equation is only applicable for steady, incompressible flows. In this equation, all the frictional effects are neglected.
We will derive Bernoulli’s equation with the help of the Conservation of Linear Momentum Principle.
There are certain restrictions in the application of Bernoulli’s Equation. For instance, we cannot apply the equation to every fluid. Obviously, there should be a certain amount of viscosity in the fluid (some fluid has less value of viscosity) and the equation cannot be implemented in every flow.
So we conclude our statement by saying that Bernoulli’s equation is only applicable to inviscid regions of the flow.
So following is the diagram that shows the regions where Bernoulli’s Equation is applicable and where it is not.
The Bernoulli’s Equation id helpful in regions outside the boundary layers where the fluid motion is governed by the combined effect of gravitational and pressure forces.
The path and motion of the particle are described by the velocity vector. In the case of steady flow, the particle passes through streamlines.
The acceleration of the fluid particle is one of the most important topics while describing Bernoulli’s equation. So following are some important key points related to the acceleration of fluid particles:
While describing the motion and path of fluid particles, it is important to describe the distance along the streamline with the radius of curvature of the streamline.
Obviously when we are explaining the distance of a fluid particle then velocity can also be related to it. But here is a condition the velocity may vary along the streamline.
In the case of defining the acceleration of fluid particles of two-dimensional flow, the acceleration is divided into two parts. One is the normal acceleration an and one is the streamwise acceleration as respectively.
The normal acceleration is due to the change in direction whereas the streamwise acceleration is due to the change in speed.
There is a condition that when the particles are moving in a straight path the normal acceleration is zero since there is no change in direction and the particle is moving in a straight line so the radius of curvature is infinity.
Bernoulli’s Equation is the result of force balance along the streamline.
We might wonder that in steady flow the acceleration is zero. But here is an interesting fact the acceleration is not zero. We define acceleration as the rate of change of velocity in time but there is no change in time. In order to exemplify this point garden hose is one of the perfect examples to describe. There is a steady flow in the garden hose and steady means no change with time but the quantity may change. When the water accelerates along the nozzle then there is a change in velocity at the inlet and outlet but at a specific point it remains constant.
So mathematically representation of velocity is as follows:
dV= ∂V∂sds+∂V∂tdt and dVdt=∂V∂sdsdt+∂V∂t
In the case of steady flow,
∂V∂t=0
And
V=V(s)
The equation of acceleration will be as follows:
as=dVdt
as=∂V∂sdsdt
as=∂V∂sV
as=VdVds
Before starting the derivation of Bernoulli’s Equation let us overview the statement of Bernoulli’s Equation and that is as follows:
The sum of kinetic, potential and flow energies of a fluid particle is constant along the streamline during the steady flow neglecting the frictional and compressibility effects.
Let us start the derivation of Bernoulli’s Equation with the theoretical part:
First of all, consider the fluid particle in a flow field with steady flow. The fluid particle is shown in the diagram as follows:
According to Newton’s Law states as
The conservation of linear momentum relation in fluid mechanics.
The fluid particle moving in the s direction along the streamlines gives the following force according to Newton’s Law:
∑Fs=mas
Here all the frictional forces and effects are neglected. The forces that are acting in the s directions are pressure forces and they are acting from both sides along with the weight of the particle.
The equations will now become:
PdA-P+dPdA-Wsin =mVdVds (1)
Here is the angle between the normal of streamline and the vertical z-axis. The value of mass is m=ρV and it will become:
m=ρdAds
The value of the weight of the fluid particle will be W=mg and it will become:
W=ρ g dA ds
The value of theta will become:
sin =dzds
We will substitute all the values in equation 1 and we will get:
-dPdA-ρg dA ds dzds= ρ dA ds VdVds
Now the next step is to cancel out the dA and we will get the following equation:
-dP-ρg dz=ρV dV
In the next step we will substitute the value of V as
VdV=12dV2
We will substitute the value in the above equation and then divide each term by density and will get the following result:
dP+12d(V2)+gdz=0
After the integration the equation for the steady flow along the streamline will be as follows:
∫dP+V22+gz=constant
In the case of incompressible and steady flow the equation will be as follows:
P+V22+gz=constant
The above equation is used in inviscid flow regions. So now Bernoulli’s equation will be as follows:
P1+V122+gz1=P2+V222+gz2
This is the final equation of Bernoulli and that is derived by the conservation of momentum that is moving along the streamline.
Bernoulli’s Equation is also viewed as the Conservation of Mechanical Energy Principle.
This principle also states that no mechanical and thermal energy is converted but they are conserved separately.
Bernoulli’s equation is implemented to an incompressible, steady flow as we have discussed earlier.
As we have also discussed that whenever the force is applied to a system the energy is transferred so we can relate that with Bernoulli’s equation which statements will; be as follows:
The work done by the pressure and the gravitational forces that are increased is equal to the increase in the kinetic energy of fluid particles.
In steady and incompressible flows, the force balance in the direction of n across the streamline is as follows:
P+∫V2Rdn +gz= constant
We have a huge discussion on the incompressible and steady flows and their derivation equation now Bernoulli’s equations in terms of compressible and unsteady flows are as follows:
∫dP+∫∂V∂tds+V22+gz=constant
Some of the important key points related to the topics are as follows:
The statement of Bernoulli’s equations says that the sum of the fluid particle,s' kinetic, potential and flow energy is constant. So the kinetic and the potential energy during the flow is easily converted to the flow energy , which changes the pressure. So density is multiplied with Bernoulli’s equation and we got the following answer.
P+ρV22+ρgz=constant
this equation has some essential terms, and one by one, I will explain each term to you.
Static Pressure
At the start of the equation, there is pressure P, and that pressure is called Static Pressure. This static pressure does not incorporate dynamic effects. Static pressure is the accurate presentation of thermodynamics pressure, and while solving the equation, we use thermodynamics tables for using the values.
Dynamics Pressure
The second term, i.e. V22, is called the dynamics pressure, and when the fluid is in motion, and there is an increase in pressure, this pressure stops isentropically.
Hydrostatic Pressure
The third term of the equation is called hydrostatic pressure, and basically, this is not the pressure because of its dependence on elevation effects. This means that this pressure accounts for the weight of fluids.
so the sum of all these pressures is called total pressure, whereas the sum of static and dynamic pressure is known as stagnation pressure, and its equation is as follows:
Pstag=P+ρV22
There are certain limitations applied while dealing with Bernoulli’s Equation. The list of constraints is as follows:
The first and foremost important limitation is that as we have derived the whole derivation of Bernoulli’s equation, you might have noticed the word steady and incompressible. So yes, Bernoulli’s equation is applicable in stable flows only. I have also mentioned the equation for unsteady flows also but there Bernoulli is not applicable.
You might have noticed one of the important factors that during the steady and incompressible flows, all the frictional and compressible effects are neglected. So yes, that is the second limitation of Bernoulli’s equation. Frictional effects in the case of short flows and large cross-sectional areas. And if the velocity is low, then that is a plus point. A slight disruption in the streamlined flow can make Bernoulli’s equation inapplicable to that flow. So much concentration and care are required for a steady and incompressible flow.
We have derived Bernoulli’s Equation from a particle moving along a streamline, so any flow in machines such as turbine fan or impeller without streamlined flow does not apply Bernoulli’s equations in flow sections. So energy equations are widely used for calculating the shaft input or output of these machines. But Bernoulli’s equation can still apply to flow sections before or past a machine.
The following restriction of Bernoulli’s equation is that Bernoulli’s equation states that density is constant, which means that the flow is incompressible. The incompressibility is applied in the case of liquids as well as gases. The Mach number should be less than 0.3 in that case. The compressibility and the density variations in the case of gases are negligible because of the low relative velocity.
Another limitation says that the increase in temperature decreases the density and vice versa. So Bernoulli’s equation is not applicable in the case where the flow section has a lot of changes in temperature (cooling and heating).
The last limitation of Bernoulli’s equation is that it is applicable in the flow streams only. As discussed above in the derivation, the final equation is applicable in the steady incompressible and flow streamline. So when the flow is disrupted or is irrotational there is no voracity in the flow field, the value of constant is the same for all the streamlines and here Bernoulli’s equation becomes applicable. The main thing is whether the flow is irrotational we don’t need to worry about the application of Bernoulli’s equation.
Bernoulli’s equations are one of the most important fields and have numerous numbers of uses in our daily life. Here I will share some of the important applications of Bernoulli’s Equation. So let us start.
One of the important applications of the Bernoulli’s Equation is a spray gun and another name is atomizer. The procedure is that we press the piston and the air is rushed from a tube that is in horizontal direction. The following diagram shows the atomizer.
So at point B, the air rushes out and the pressure is decreased from p2 to p1. Because of the pressure decrease, the liquid is raised in vertical tube A. So the high air pressure collides with the liquid rise tube and results in fine sprays. This fine spray is used in many different sprays, Bunsen burners, perfumes and many more sprays.
Who does not like travelling on aeroplanes? But do you know how aeroplanes take off and land? let me explain to you the theory of rising aeroplanes. Actually, the aeroplane rise as a result of the lift force that is acting on the wings. The lift is a result of an object placed asymmetrically and that is moving through the fluid. Bernoulli’s equation says that the increase or decrease in the velocity affects the pressure and causes variations in it. So pressure variations create an upwards force and that upward force is shown in the following diagram.
This is also one of the well-known examples of Bernoulli’s equation. A Venturimeter is used for the measurement of flow speeds in the pipe that is not uniform. As we know that Bernoulli’s equation is applied in narrow shallow and wide pipes where the height is the same and the area at the inlet is greater than the outlet and the velocity at the outlet is greater than the inlet velocity and at last, the pressure at the outlet is greater than the pressure at the inlet. so there is a collectively force applied when the fluid enters the throat and the force slows down on leaving. The following diagram shows the working.
Now we will move towards our last topic of this article that is about the Energy Equation. This topic is also one of the fundamental topics of fluid dynamics. So without wasting any time let us start the topic and see the equations of it.
In this article, I will cover one of the essential topics of fluid mechanics as the title shows three crucial topics—the first concerns the conservation of mass and how it plays a critical role in our daily life. The next one involves the conservation of kinetic and potential energy and is all about the flow energy of a fluid stream. The Bernoulli equation is all about considering the energies, handling them of fluid, and neglecting the viscous forces. The last topic is energy equations, which is about the conservation of energy principle. As we know that while dealing with energy in fluid mechanics, the mechanical energy can be separated from the thermal energy and end, considering it an energy loss.
So without wasting time, let us start the article with the mass.
I will start with the laws of conservation. As you have heard about many laws, while discussing things related to mass, it is essential to examine the conservation laws. So let us discuss conservation laws one by one.
The definition of the law of conservation of mass is as follows:
In any close or isolated system, matter cannot be created nor destroyed but can be changed and conserved.
Some of the essential key points related to the law of conservation of mass are as follows:
In the system, the mass remains constant throughout, and the according to the equation it says:
msys=constant
dmsysdt = 0
Here, dmsysdt is the mass change r within the system boundaries.
so, in the case of the control volume, the conservation of mass in terms of rate form is presented as follows:
min-mout=dmcvdt
here, the min is the total rate of control volume;l volume, and similarly, mout is the mass flow rate out of control volume
Also, dmcvdt is the mass change rate within the control volume.
You might have heard that the conservation of mass is interrelated with the continuity equation. So the conservation of mass defined in terms of the differential control volume is called Continuity Equation.
The definition of the law of conservation of momentum is as follows:
The Law of Conservation of Momentum states that the momentum of an isolated system is constant.
Some of the essential key points related to the law of conservation of momentum are as follows:
In fluid dynamics, the conservation of momentum is interrelated to Newton’s Law of motion. In solid mechanics, Newton’s Laws of motion are implemented.
As we know, that momentum formula is about the product of mass and velocity.
Newton’s second law of motion says:
The rate of change of momentum of the body is equal to net forces acting on it.
The statement of conservation of energy is as follows:
The net energy transfers to or from the system during a process be equal to the change in energy content of the system.
Some of the essential key points related to the conservation of energy are as follows:
As we know that energy cannot be created nor destroyed, but it can be transformed from one state to another. So the statement of Conservation of energy is similar as it says that the energy transferred to the system is equal to the change in the system's content. So it means that no energy is destroyed it is transformed into another form.
The energy balance equation of conservation of energy will be:
Ein-Eout=dEcvdt
Here, Ein and Eout are the total energy in and out of the control volume.
dEcvdt is the rate of change energy within the control volume.
It is the first part of the article, and while discussing this topic, we should know that it is one of the most fundamental principles in nature. At the start of the article, I have given you a brief introduction about the conservation of mass, the statement, and the energy balance equation.
Now I will discuss those key points not discussed before in detail. So their explanation is as follows:
The mass has a much similar property as the energy. One of the significant statements is that it cannot be created or destroyed. But mass can be converted to energy and vice versa with the help of the following statement and i.e.
E=mc2
Yes, you guessed it right. This is Einstein Equation. Where c is the speed of light, and the value is c=2.9979108m/s.
The equation helps us detect the change in mass whenever there is a change in energy.
As the statement of the conservation of mass shows that the system’s mass remains constant but in the case of a closed system. While discussing the control volumes, there might be some changes. The significant change will be that mass can cross boundaries.
These two words are well known in fluid mechanics, so readers should know these two when discussing the conservation of mass. So the definition of mass flow rate is as follows:
The mass flow through any cross-section per unit time is called mass flow rate.
Some of the essential key points related to the mass flow rate are as follows:
The fluid passing through any cross-sectional area has a flow rate. Let us suppose it passes through some small area; then, the differential mass flow rate will be:dAc.
Then, the equation will be like this:
m=ρVndAc
Here, is used for quantities such as heat, work, and mass transfer, whereas d ( differential quantity) is used for the properties.
Let us have the flow of fluid passing through some annulus with inner radius as r1 and outer radius r2 as then the equation will be:
12dAc=Ac2-Ac1=π(r22-r12)
Also, we know that the total mass flow rate is as follows:
12m=mtotal
After the integration, the mass flow rate equation will be as follows:
m=∫Ac δm = ∫Ac ρVndAc
The equation is very informative but is not used in daily life because of the integration. So the flow rates are values taken as average. The importance of density and velocity varies accordingly to the size of the pipe. Due to non-slip conditions, the value of velocity varies
So the equation of average velocity is as follows
Vavg=1Ac∫AcVndAc
In the case of compressible and incompressible flows when the density is uniform throughout the cross-sectional area, then the equation will be:
m=ρVavgAc
The definition of volume flow rate is as follows:
The volume of fluid flowing through any cross-section per unit of time is known as the volume flow rate.
Some of the important critical points related to the volume flow rate are as follows:
The equation of volume flow rate is as follows:
V=∫AcVndAc
V=VavgAc
V=VAc
One of the essential points that should be kept while dealing with the volume flow rate is that it should not be confused with the symbol Q as it also presents the flow rate.
There is a relation between the mass and volume flow rate, and that relation is explained by the equation.
m=ρV=Vv
The statement of conservation of mass principle is as follows:
The newt mass transfer to or from a control volume during the interval of time is equal to the net change in total mass within the boundaries of the control volume.
The theoretical equation is as follows:
(Total mass entering the CV during ∆t) – (Total mass leaving CV during ∆t)
=
(Net change in mass within the Control Volume during ∆t)
The mathematical equation will be as follows:
min-mout=∆mcv
This equation can also be expressed in terms of rate form:
min-mout=dmcvdt
Here one of the essential vita points that must be kept in mind is that the conservation of mass principle is performed within the control volume boundaries.
The theory and the mathematical equations about the conservation of mass principle are as follows:
First of all, we will consider the control volume of irregular shape as shown in the diagram below:
So the mass of differential volume that is in the control volume will be:
dm=ρdV
The equation of total mass within the control volume will be:
mcv=∫cvdV
Now I will show you the equation of the rate of change of mass within the boundary of a control volume:
dmcvdt=ddt∫cvρdV
The conservation of mass principle has a special case in which no mass can cross the boundary. We can conclude that control volume restrictions are much similar to the closed system. The equation of conservation of mass principle, in this case, will be as follows:
dmcvdt=0
Now we will further assume that the mass is flowing from the differential area of the control volume. Here the unit vector will be n of differential area (dA). V will be the flow velocity and this whole theory is presented in the diagram that is mentioned above.
There is an angle at a point where the velocity can intersect the differential area.
The mass flow rate here is proportional to the normal component of velocity as:
Vn=Vcos
here there are some values of theta that are applied to the above-mentioned equation. Such as =0 when the outflow is maximum and the flow is normal to the differential area. The value of =90° when the flow is tangent to the differential area and the outflow is minimal to zero. The value of =180 when inflow is maximum and the flow and differential area is opposite to each other also the flow is normal to the differential area.
So the equation of the normal component of velocity will be as follows:
Vn=Vcos =V.n
As this is the important principle of fluid mechanics, it comprises equation many. Now there is the equation of differential mass flow rate in which the mass flow rate passing through the differential area is directly proportional to the fluid density and it is normal to the flow area:
m=ρVndA=ρVcos dA=ρV.ndA
Now there is an equation of net mass flow rate and it is an overall equation of control volume in which the flow rate moves in and out through the control surface. So the equation is as follows:
mnet=∫csm=∫csVndA=∫csρ(V.n)dA
Here are some important values of theta as we know that V.n = Vcos so the value will be <90° in the case of outflow and it will be >90° in the case of inflow. So whenever the value of the net mass flow rate is positive, it shows that it is an outflow; whenever the value is negative, it shows that the net flow is inflow.
Now there was an equation at the start which was min-mout=dmcvdt
We will rearrange it and will present it as an equation of general conservation of mass equation and it is as follows:
ddt∫cvρdV+∫csρ(V.n)dA=0
For your convenience I will explain the above equation to you. The above equation says that the rate of change of mass within the boundary of control volume along with the net mass flow rate through the control surface is said to be equal to zero.
We can also derive the equation with the help of Reynold’s Transport Theorem.
Now I will show you our second last equation and this is associated with the above equation. What we will do is we will spill the above equation into two main parts.
One part comprises the outgoing flow streams and the value will be positive. The second part comprises the incoming streams and the value will be negative. So the equation will be as follows:
ddt∫cvρdV+t∫AVndA-in∫AVndA = 0
Here A is the area of outlet and inlet and the summation value shows all the collective values of inlet and outlet.
Now moving toward, the last equation will be the end of this topic. In the last equation I will show you how mass flow rate can be shown in some different equations:
ddt∫cvρdV=inm -outm
or
dmcvdt=inm -outm
This topic also comes under the conservation of mass of principle. So the essential key points are as follows:
Whenever there is a steady flow, the total mass within the boundary of the control volume remains constant.
But when we have studied the conservation of mass principle it says that the mass that is entering the control volume should be equal to the mass leaving it.
Toto explain further I will unquote an example that says that while watering our flowers with the help of a garden hose. The water passing through the nozzle of the garden hose is moving in a steady motion. So the water entering the nozzle per unit time equals the water leaving per unit time.
While dealing with the steady flow, the amount of mass is not considered rather the mass flow per unit time is always considered. And that amount of mass flowing per unit time is the mass flow rate.
In the case of steady flows, the flow's inlet and outlet are always considered equal. Such as shown in the diagram.
inm =outm
There are numerous mechanical devices in which a single stream with one outlet and inlet are used. Examples are turbines, diffusers and pumps. So to have an equation for them to calculate the steady flow the equation will be as follows:
m1=m2 1V1A1=2V2A2
Here 1present the inlet of flow and 2 presents the outlet of flow.
Now here comes the last topic of conservation of mass principle in this topic we observe a special case study. So without wasting any time let’s have a look at the important key points:
In the case of liquids, when the fluid is compressible the conservation of mass has a defined line.
The equation for the steady and the incompressible flow will be as follows:
in =out
Here the density is cancelled from the side. So in the case of single stream, steady and incompressible flow, the equation will be as follows:
1=2=V1A1=V2A2
We have a detailed explanation of the conservation of mass but after this equation, you might have questions in your mind about the conservation of volume. So for your information, there is no such thing exists as the conservation of volume.
In the case of steady flow, the volume flow rate at the inlet and the outlets are different and may vary from device to device.
To clarify this point let us take an example of an air compressor. The volume flow rate at the outlet is less as compared to the inlet. As the mass flow rate at the inlet and outlet is constant. The reason for this is that there is a high density of air at the exit of the compressor.
In the case of steady flow, the mass flow rate along with the volume flow rate is the same or constant throughout the whole process.
so that is a whole detailed discussion about the first part of this article which covers almost every domain. So now I will start the second part of the article, which is about the Bernoulli Equation. Without wasting any time let us start.
Before starting this important topic there are some of the important terminologies related to the theorem that are frequently used throughout. So let us have a look at them.
Some important key points related to the system are as follows:
In thermodynamics, a system is a quantity of matter of fixed quantity.
The system is mostly closed. system. There is a change in the size and shape of the system but due to the closed system, no mass can cross the boundary.
One of the important points is that the mass of the system remains constant.
Some important key points related to the control volume are as follows:
The control volume is also known as the open system and it is defined as the region that is specified for a case study.
In the case of a control, volume mass can cross and move in and out of the boundary and which is called the control surface.
One of the important key points related to the control volume is that the mass of the control volume decreases during the process but, the volume remains constant.
The important key points related to Reynold’s Transport Theorem are as follows:
In fluid mechanics, control volumes are easy to handle and many of the laws and theorem are based on it.
So Reynold’s Transport theorem is actually a relationship between the rate of change of extensive properties of the system and control volumes.
A system is considered arbitrary when Reynold’s Theorem is being considered. But here the derivation is also being carried away.
Reynold’s Transport Theorem provides a relationship between the system and control volume.
Let us start the theorem with a background of it:
First of all, we will consider the flow that is moving from left to right and that flow is passing through an expanding field.
As the fluid flow the upper and the lower bounds of the fluid flow are considered to be streamlines of flow. Between any two streamlines, it is assumed automatically that the flow will be uniform and so in this situation, it is also considered.
So there is a diagram that is mentioned below that shows the moving system and control volume (shaded region) of the flow field is considered at the times t and ∆t. the streamlines are also shown.
Here one interesting fact is that the system and the control volume are the same as the system seems to be coinciding with the control volume.
With reference to the diagram, in part (1) the system moves with uniform speed V1 and so in part (2) it moves with speed V2 uniformly.
If you look at the diagram, then there is a central region called the hatched region.
The region that is uncovered by the system during the motion is mentioned as section I and this is part of the control volume. but the new region that is covered by the system is said to be section II and this is not part of the control volume.
Now let us start the derivation. For derivation let us suppose B as extensive property and that may be mass, energy or momentum.
Moreover, let us suppose
b= B/m
Now this b=B/m is the intensive property (we are supposing it).
So the extensive property is additive, they can be added easily.
so the extensive property at times t and ∆t are presented as follows:
Bsys,t=BCV,t
Bsys,t+∆t=BCV,t+∆t - BI,t+∆t+BII,t+∆t
Now we will get to the final result by taking two steps. The first step will be subtracting equation 1 with equation 2 and then taking the limit ∆t→ 0. So the final result will be:
dBsysdt=dBCVdt-Bin+Bout
The value Bin and Bout is as follows:
Bin= b11V1A1
Bout= b22V2A2
Here A1 and A2 are the cross-sectional areas.
And the values of BI, t+∆t and BII, t+∆t is as follows:
BI, t+∆t = b11V1∆tA1
BII, t+∆t=b22V2∆tA2
Now we are on the second last step and it is as follows:
Bin=BI=b11V1A1( after taking limit)
Bout=BII=b22V2A2 (after taking the limit)
The important point here is that property is the extensive property and the time rate change of the system is equal to the time rate of change of B of control volume along with the net flux B of control volume by mass.
Now the final result will be:
Bnet=Bout-Bin=∫CS ρbV.n dA (inflow is negative)
So below mentioned is the diagram that shows the whole working of the derivation:
In the case of the control volume, the equation will be as follows
BCV=∫CV ρb dV
Here the following equation presents the time rate of change of property B content of the control volume:
dBCVdt= d/dt∫CV ρb dV
so the system to control volume in the case of the fixed control system will be as follows:
dBCVdt= ddt∫CV ρb dV +∫CS ρbV.n dA
So that is the explanation related to Reynold’s Transport Theorem.
I hope you have learned a lot through this article. Thank you for reading.
Flow visualization widely uses Computational Fluid Dynamics (CFD) and physical experiments. So following are the types of patterns that can be visualized computationally and experimentally. So without wasting any time, let us start.
There are two patterns. I will first explain the streamlines. So the definition of streamlines is as follows:
A streamline is a curve that is tangent everywhere to the instantaneous local velocity vector.
You might not understand the definition by reading it, so just for your ease, let me explain to you in few key points:
A streamline defines fluid’s motion throughout the flow field.
A streamline acts as an indicator of the instantaneous direction of motion of the fluid. To explain this situation, let me exemplify it with an example.
For instance, when we throw water on a solid surface, we observe the fluid flow pattern on the wall. That pattern is a streamlined pan in which the water is separated, moved in recirculating motion, or coming off the wall.
There is an equation of streamline; let me explain the background of the equation to you. Before that, have a look at the diagram.
As you can see, there is an infinitesimal arc along the streamline. Here the infinitesimal arc length is as follows:
dr=dxi+dyj +dzk
Here is a condition that dr should be parallel to the local velocity, whose equation is as follows:
V=ui+vj +wk
By utilising the geometric rules, infinitesimal arc dr is proportional to the local velocity, and the equation of streamline will be like this:
drV=dxu=dyv=dzw
Here, dr is the magnitude of infinitesimal arc length, and V is the magnitude of velocity.
To obtain the equation in an (x, y) plane, we will integrate equation 1 and get the equation of streamline in an (x, y) plane. The equation is as follows:
(dydx)along astreamline=vu
Now I will give a brief explanation about the Streamtubes. So the definition of Streamtubes is as follows:
A Streamtubes is a bundle of streamlines similar to the communication cable with optic fibre cables.
You might not understand the definition by reading it, so just for your ease, let me explain to you in few key points:
As I have discussed earlier, streamlines are parallel to the local velocity, so according to the theoretical information, the fluid cannot cross the streamlines.
A simple diagram elaborates on tee difference between the streamlines and streamtubes.
The definition of Pathlines is as follows:
A Pathline is an actual path travelled by an individual particle (obviously a fluid particle) at some time period.
The following are some essential key points related to the pathlines:
You might have noticed one thing while reading the definition of pathline and Lagrangian are much similar. Both follow a path of an individual particle when the fluid flows.
There is a technique named Particle Image Velocimetry (PIV) that is used to measure the velocity field in one flow of a specific plane.
Let me explain to you the PIV technique briefly. So what happens in PIV is that those small particles are released in fluid, as shown in the diagram.
Then, the flow is observed by a two-flash light to make two spots on a film of every small particle moving. The magnitude and direction of the velocity of each particle location are fixed because the particle size is small. In today’s modern era of science and technology, many modern computers and digital photography have enabled this feature.
In order to trace the location of particles by the following equation:
x=xstart+tstarttV dt
So, the diagram mentioned earlier shows the pathline following the actual path of fluid particles.
There is a condition that if the velocity field is steady, the fluid particles are bound to follow the streamlines.
The definition of a streakline is as follows:
A streakline is the locus of fluid particles passed sequentially through the flow’s specific (prescribed) point.
The following are some important key points related to the streaklines:
The streaklines are one of the most common flow patterns produced through physical experiments. To explain this point more clearly, let me explain to you through an example. If we insert a tube (specifically in small size) in the flow and then add a continuous stream of tracer fluid, the pattern produced due to the addition will be streaklines. The flowing diagram shows the streaklines produced when the constant stream of tracer fluid (colored fluid) is added to the flow. The fact about the diagram is that the streaklines are similar to the pathline and streamlines.
Here is an interesting point to be noted: if the flow is steady, then the pathlines, streamline, and streaklines are all identical.
So you might be confused by the statement that pathlines, streamlines, and streaklines are similar in steady flow. But in the case of unsteady flow, the scenario is entirely different.
There is a difference, i.e., streamlines give an instantons flow pattern (by definition) but in the case of streaklines is an instantons snapshot of time-integrated flow patterns. And the pathlines are the time-exposed flow path of an individual particle at some time.
The equation to find the integrated tracer particle is as follows:
x=xinjection+tinjecttpresentV dt
The definition of timelines is as follows:
A timeline is a set of adjacent fluid particles that were marked at the same time.
The following are some key points related to the timeline:
The fluid flows in which there is uniformity and a steady flow, then timelines are observed.
Practically timelines can be generated in any water channel with the help of hydrogen bubble wire.
As we all know, flow properties vary from time to time, and in space, it is necessary to plot flow data in various ways. In this article, I am going to explain three kinds of plots that are profile plots, vector plots, and contour plots, respectively.
So without wasting any time, let’s start explaining the plots.
The definition of a profile plot is as follows:
A profile plot indicates how the scalar property varies along some desired direction in the flow field.
The following are some essential key points related to profile plots:
It is one of the simplest plots much similar to the XY-plots.
As the definition describe that the profile plots are created for the scalar quantities, but in fluid mechanics, velocity profile plots are used. As velocity is a vector quantity. So, to create a profile plot, we either use magnitude or velocity.
The definition of the vector plots is as follows:
A vector plot is an array of arrows reflecting the magnitude along with the direction of vector quantity at an instant of time.
The following are some essential points related to the vector plots:
Streamlines are used to present the direction of the instantaneous velocity; here, they do not show the velocity magnitude.
A flow pattern is a vector plot for experimental and computational fluid flow. They have an array of arrows that indicate both magnitude and direction of a vector quantity.
The following definition of a contour plot is as follows:
A contour plot shows curves of constant values of scalar property (or magnitude of a vector property) at an instant in time.
The following are some essential key points related to the contour plots:
The contour plot may have curves indicating various properties called Contour Line Plot.
Some of the contours are filled with color of grey are called Filled Contour Plot.
In fluid mechanics, the elements have four fundamental types of motion or deformation in two dimensional as follows. It is interesting to know the fact that all four of these motions can act at the same time. Yes, you heard it right. Isn’t it amazing? In fluid dynamics, the motion and deformation of liquid elements at different times are described. So the deformation rates are expressed in terms of velocity and derivatives of velocity.
Translation
Rotation
Linear Strain (Extensional Strain)
Shear Strain
So let us start explaining one by one.
The translator and rotatory motion are one of the most common motions that are observed in our daily life. For the three dimensions, a vector is used to define the translator rate. In Cartesian coordinates the rate of translation is as follows:
V=ui+vj+wk
The rate of rotation or the angular velocity is defined to be the average rotation rate of two perpendicular lines that intersects at some point.
Example:
In order to exemplify this whole situation, let us explain through some examples.
Let us take an initially square fluid element and consider its bottttttttt.
Technically, the left and the bottom edge of the element intersect at some point. Thus we can say that they are perpendicular to each other.
Now, these two lines tend to rotate in a counterclockwise direction (which is said to be the positive direction).
Toto, show a clearer picture to you let me show you a diagram. So the below-mentioned diagram explains the rotational effect.
One of the main points to observe is that the angle between the two lines remains the same, which is 90 degrees.
So the line rotates at the same rate of rotation.
Here there is also one of the most essential points about the rate of rotation is that when the case is two dimensional then the fluid elements move in translator motion and then rotate. But while rotating, they deform easily.
In order to calculate the rate of rotation, the whole thing is calculated by the scenarios mentioned above, i.e. two lines are taken named a and b respectively. These two lines intersect at a specific point called P ( as mentioned above that this procedure is in two dimensions. Which Means that they are in the XY plane).
These lines are being followed and rotated in an infinitesimal increment of time, and that is:
dt=t2-t1
Line a rotates at some angle aand line b also rotates at a specific angle b. The average rotation angle will be:
a+b2
So the final equation will be as follows:
ω=ddt(a+b2)
ω=12(∂v∂x-∂u∂y)
The definition of linear strain rate is as follows:
Linear strain rate is defined as the rate of increase in length per unit length.
Some of the important key points related to the linear strain rate are as follows:
The linear strain rate depends upon the direction of the line segment and this line segment measures the linear strain.
But here important point should be kept in mind i.e. that the linear strain rate cannot be defined as vector or scalar quantity.
The linear strain rate can be defined in an arbitrary direction.
The linear strain rate can be defined in Cartesian coordinates by the formula as follows:
xx=∂u∂x yy=∂v∂y zz=∂w∂z
The definition of the volumetric strain rate is as follows:
The rate of increase in the volume of fluid element per unit volume is called the volumetric strain rate.
Some of the important critical points related to the volumetric strain rate are as follows:
In an incompressible flow, the volumetric strain rate is zero.
With the increase in volume, the kinematic property is always positive.
The other definition word for the volumetric strain rate is called Rate of Volumetric Dilatation.
The rate of volumetric dilatation is remembered by an example. Let us take an example of the iris of expands enlarging when there is less light.
The formula of volumetric strain rate in Cartesian coordinates is as follows:
1VDVDt=1VdVdt=xx+yy+zz=∂u∂x+∂v∂y+∂w∂z
The definition of the shear strain rate is as follows:
A shear strain rate is defined at the point as half of the rate of decrease of the angle between two initially perpendicular lines that intersects at some points.
Some of the important key points related to the shear strain rate are as follows:
In order to explain the definition, let us explain through some examples. Let us see a diagram first that is mentioned below:
In this diagram, the angle is at first 90 in the lower left corner and upper right corner of the element of fluid the angle decreases so that is a positive shear strain. But the angles at the upper-left and lower-right square fluid element increase so that is negative shear strain.
In the Cartesian coordinates system, the shear strain rate will be:
xy=12∂u∂x+∂v∂x zx=12∂w∂x+∂u∂z yz=12∂v∂z+∂w∂y
I hope you have learned a lot through this article. Thank you for reading.
Hello friend. In this article, I will cover essential points related to fluid kinematics, i.e., what fluid kinematics is and how the motion of the fluids can be explained without describing the forces acting on them. Further, I will explain Lagrangian and Euler about the motion of fluid and a lot more. So buckle up, and let’s start.
The definition of fluid kinematics is as follows:
Fluid kinematics is related to fluid motion without considering the forces responsible for the motion.
The following are some essential key points related to fluid kinematics:
The nature of fluid motion is categorized into two types:
I will extensively explain both types in upcoming topics.
The acceleration, velocity, flow rate, and nature of fluid flow are considered while working on fluid kinematics.
First of all, I will explain the Lagrangian. So the definition of Lagrangian is as follows:
Lagrangian Description of fluid flow deals with individual particles and their nature, and the working trajectory of each particle is calculated separately.
And the definition of Euler is as follows:
Euler’s Description of fluid flow deals with the concentration of the particles, and the number of particles and diffusion are all calculated.
The following are some essential key points related to the Lagrangian Description:
Lagrangian is all about tracking the path of individual particles.
In this regard, Newton’s Laws are beneficial in detecting the pathways. With the help of their kinetic energy, speed, distance velocity, and acceleration.
As we all know, fluid is Continuum in nature. If you are following my fluid mechanic’s articles series, you have forwarded to this word, where I have extensively described the continuum. So it becomes pretty challenging to follow the path of fluids as they deform whenever a direction changes.
In the Lagrangian description, the position and velocity of individual particles must be tracked throughout.
To describe through a simple example the balls on the snooker table. In this case, each ball is responsible for its path.
The following are some essential key points related to the Euler Description:
In the case of Euler’s description, the control volume is defined through which the fluid can move in.
In this description method, there is no need to track the velocity and position of fluid particles.
In this description, space and time are defined within the domain of control volumes.
As I have defined in previous articles, pressure is a scalar quantity. So pressure field in case of three-dimensional fluid flow in Cartesian co-ordinates can be defined as follows:
P = P (x, y, z, t)
The velocity field is mentioned as follows:
V=V(x, y, z, t)
The acceleration field is as follows:
a= a (x, y, z, t)
The interesting fact to know is these three field(along with some others but for now only these three fields) variables define the flow field:
V=(u, v, w)
V=ux,y,z,ti +vx,y, z, tj +w(x, y,z,t)k
I will explain it through a simple example to depict the Lagrangian and Euler description.
The example is as simple as one person standing beside the river. So when he goes through the probe, which will move downstream with water, this situation will be classified as Lagrangian. This will be classified as Euler’s description when he anchors the probe in some fixed location in the water.
The applications of fluid Kinematics are as follows:
The purpose of dealing with Fluid kinematics is that fluid has excellent properties, and they are used almost in every vehicle. So for lubrication purposes and as driving fuel.
Anything that moves has kinetic and potential energy and fluids. They are used in hydroelectric power plants for the generation of electricity.
In refrigerators and air conditioners, refrigerants used are fluids. The primary purpose of refrigerants is to absorb the heat (from the room or, in the case of refrigerators, anything that is kept inside them) and keep them cool, then release the heat into the atmosphere. In the case of air conditioners, heat is absorbed from the room to keep them cool and release that heat into the atmosphere.
Surprisingly fluids can be used as a renewable energy resource. One of the simplest and best examples is the water used in tidal power plants for electricity generation. Moreover, a vegetable oil known as biodiesel is used in many vehicles. Wind and air are also used as renewable energy sources.
One of the essential resources of electricity in thermal power plants. In thermal power plants, water (used as fluids) is heated to form steam, which then turns on the turbine. The turbine then turns on the generator, which generates electricity. So the purpose of explaining this whole procedure is to explain the importance of fluid in electricity generation.
So that are the applications of fluid kinematics in our daily life. Dear friends, I will explain two significant ways to describe motion.