Unit - 2
Types of Fluid Flows
Q1) Explain Steady and Unsteady flow?
A1)
Steady flow –
- The type of flow in which the fluid characteristics like velocity, pressure, density, etc. at a point do not change with time is called steady flow.
- Example: Flow through a prismatic or non-prismatic conduct at a constant flow rate Q m3/s is steady.
- Mathematically, u/ t = 0, w/ t = 0, p/ t = 0, rho / t = 0
Unsteady flow:
- It is that type of flow in which the velocity, pressure or density at a point change w.r.t. Time.
- E.g.: the flow in a pipe whose value is being opened or closed gradually.
- Mathematically, u/ t not = to 0, w/ t not = to 0, p/ t not = to 0, / t not = to 0
Fig.: Steady and Unsteady flow
Q2) Explain Uniform and Non-uniform flow?
A2)
Uniform flow –
- The type of flow, in which the velocity at any given time does not change with respect to space is called uniform flow.
- E.g.: Flow through a straight prismatic conduit.
- Mathematically, V / s = 0 when t = constant
Non-uniform flow –
- It is that type of flow in which the velocity at any given time changes with respect to space.
- E.g.: Flow through non-prismatic conduit.
- Mathematically, V/ s not = to 0 when t = constant
Fig.: Uniform and non-uniform flow
Q3) Explain Laminar and Turbulent Flow?
A3)
Laminar Flow –
- A Laminar flow is one in which paths taken by the individual particles do not cross one another and move along well-defined path.
- E.g.: Flow of blood in veins and arteries.
Turbulent Flow –
- A turbulent flow is that flow in which fluid particles move in a zig zag way.
- E.g.: High velocity flows in conduit of large size.
Fig.: Laminar and turbulent flow
Q4) Explain Rotational and Irrotational flow?
A4)
Rotational Flow –
- A flow said to be rotational if the fluid particles while moving in the direction of flow rotate about their mass centers.
- E.g.; Motion of liquid in a rotating tank.
Irrotational flow –
- A flow said to be rotational if the fluid particles while moving in the direction of flow do not rotate about their mass centers.
- E.g.: Flow above a drain hole of a stationary tank or a water basin.
Fig.: Rotational and irrotational flow
Q5) What is Compressible and Incompressible flow?
A5)
Compressible Flow –
- It is that type of Flow in Which the density () of the fluid changes from point to point.
- E.g.: Flow of gases through orifices, nozzles, gas turbines, etc.
- Mathematically, not = to constant
Incompressible Flow –
- It is that type of flow in which density is constant for the fluid flow. Liquids are generally considered flowing incompressible.
- E.g.: subsonic aerodynamics.
- Mathematically, = constant
Fig.: Compressible and incompressible flow
Q6) Explain Subcritical, Critical and Supercritical flows?
A6)
Subcritical Flow:
- When Froude number is less than one or V> ,the flow is said to be subcritical flow or tranquil or streaming flow.
Fr <1
Critical flow:
- When Froude number is less than one or V= ,the flow is said to be Critical flow.
Fr =1
Supercritical Flow:
- When the Froude number is greater than one or V>, the flow is said to be super critical flow or Rapid flow or Shooting flow.
Fr >1
Q7) What is One, Two and Three Dimensional flows?
A7)
One dimensional flow –
- It is that type of flow in which the flow parameter such as velocity is a function of time and one space coordinate.
- E.g.: Flow in a pipe where average flow parameters are considered for analysis.
- Mathematically, u = f(x), v = 0 & w = 0
Two-dimensional Flow –
- The flow in which the velocity is a function of time and two rectangular space coordinates is called two-dimensional flow.
- E.g.: Flow between parallel plates of infinite extent.
- Mathematically, u = f1(x, y) v = f2(x, y) & w = 0
Three-dimensional flow –
- It is that type of flow in which the velocity is a function of time and three mutually perpendicular directions.
- E.g.: Flow in a converging or diverging pipe or channel.
- Mathematically, u = f1(x, y, z) v = f2(x, y, z) & w = f3(x, y, z)
Fig.: One two- and three-dimensional flow
Q8) What is Streamline?
A8)
- A streamline may be defined on as an imaginary line within the flow so that the tangent at any point on it indicates the velocity at that point.
- An important concept in aerodynamics research touches on the concept of simple methods. The arrangement is the process followed by the weightless particles as they flow and flow.
- It is easier to visualize a streamline when moving through the body (unlike walking and flow).
- The integrated broadcast around the airfoil and near the cylinder. In both cases, we move with the object and the flow continues from left to right.
- As the streamline is followed by moving particles, at all points along the path the velocity is moving along the path. Since there is no standard object on the road, the magnitude cannot cross a straight line.
- The complexity contained between any two streams remains the same throughout the flow. We can use Bernoulli's figure to associate the pressure and speed associated with easy movement.
- Since there is no weight passing over the airfoil (or cylinder), the surface of the object is the easiest way.
- The planes fly through almost dry air, but we build and test them using air channels, where the model of the plane is stopped and the wind is blown through that model.
- The idea of keeping the airfoil fixed and that the airflow passing through the airfoil can be a little confusing. However, you experience the same kind of thing every day
- If you stand on a corner and watch the car pass by, the atmosphere is a little quieter. As the car moves in the air there is aerodynamic force present.
- Now if you were in a car and you put your hand out the window, you could feel the aerodynamic force pushing your hand. It sounds like air is passing through your hand as fast as the car is moving.
- The power of the car is the same, whether you are standing in a corner or riding in a car. The same is true of airplanes. Whether the plane is moving in the air, or the wind is being pushed past the plane, the force is exactly the same.
- It is usually simple, inexpensive, and (in some cases) harmless to check planes in the air tunnel before attempting to fly.
- Streamlines are a family of curves that quickly turn into velocity vector flow. This indicates the direction in which the waterless object will move at any time.
- Stream line, In Liquid Mechanics, a method of synthetic particles suspended in a liquid and traveled with it.
- With a constant flow, the fluid flows but straight lines are arranged. Where the dispersion meets, the velocity of the liquid is very high; when they open, the liquid is quiet. See also laminar flow, turbulent flow.
Fig.: Stream line
Q9) The velocity of a flow is given by v = -x i + 2y j + (5 - 2) k. Derive the equation of streamline passing through a point (2, 1, 1).
A9)
u = -x, v = 2y and w = 5-z
The streamline,put the value of u, v and w
Consider,
Stream line passing through x=2, y=1, z= 1
Now consider,
Stream line passing through x=2, z=1
Q10) For the three-dimensional flow, the velocity distribution is given by u-x, v-4-2y, w z-2. What is the stream line equation passing through (1.-2, 3)?
A10)
The streamline,
Put the value of u, v and w
Consider
By integrating
Stream line passing through x=1 y= =-2
Now considering
From equation
Q11) The velocity component in a steady flow is u=2kx, v= 2ky, w = - 4kz. What is the through the point (1, 0, 1).
A11)
The streamline,
Put the value of u, v and w
Now considering
Stream line passing through x = 1, z=3,
This is the equation of streamline passing through the point (1,0,1)
Q12) The velocity field for a two-dimensional flow is given by V= 4x³i-12x²yj. What is the equation of streamlines?
A12)
Velocity vector v = 4x'i-12x²yj
Velocity component u = 4x², v = -12x²y
Stream line
By integrating,
3 log x = -log y + log C
The curve defined by x³y = C₁, C₂, C, ...etc.
Q13) What is Path line?
A13)
- Path lines are the trajectories that are followed by the particles of each body. This can be thought of as "recording" the method of a liquid object in a certain flow of time. The direction of the path to be determined will be determined by the fluid flow of the liquid at each time point.
- Path Lines - A line path is a line followed by water particles as they travel over a period of time. Thus, the line of motion indicates the direction of the velocity of the same particle of liquid at successive times. While the formulation shows the speed direction of many liquid particles at the same time.
- The liquid particle always moves the tangent to the grid, so, with a constant flow, the lines of direction and direction are the same.
Fig.: Path lines
Q14) Explain Three-dimensional continuity equation?
A14)
- Considers a fluid element (control volume)- parallelepiped with sides dx, dy and dz as shown in fig.
Fig.: continuity equation
Let, = Mass density of the fluid at a particular instant
u, v, w= components of velocity of flow entering the three faces of the parallelepiped.
Rate of Mass of fluid entering the face ABCD
= X velocity in x-direction X area of ABCD
= u dy dz
Rate of Mass of fluid leaving the lace FEGH
= u dy dz +
:. Mass accelerated per unit time, due to flow in x-direction
= u dy dz – [ u+ dx] dydz
= - - ( u) dx dy dz
Similarly, the gain in fluid mass per unit time in the parallelepiped due to flow in Y and Z- direction.
= ( v) d x d y d z ……… (in Y- direction)
= ( w) d x d y d z ……… (in Z- direction)
The, total gain in fluid mass per unit for fluid along three co-ordinate axes
= -[ ( u) + ( v) + ( w)] dx dy dz …………1
Rate of change of mass of the parallelepiped (control volume)
= ( d x, d y d z) ………………….. 2
From Equation 1& 2
-[ ( u) + ( v) + ( w)] dx dy dz = ( dx dy dz)
Simplification and rearrangement of teams would reduce the above expression to
( u) + ( v) + ( w) + =0
This eQ is the general equation of continuity in three dimension and is applicable to any type of flow and for any fluid whether compressible as incompressible
For steady flow (=0) incompressible fluids ( = constant) the equation reduces to
++=0
Q15) The velocity components for a two-dimensional incompressible flow is given by
A15)
1) u = 4 xy + y². v = 6 xy + 3x
2)
Check whether the flow satisfies continuity?
A15)
(i) u = 4 xy + y², v = 6 xy + 3x
Differentiating
For two-dimensional flow continuous equation is
Velocity components does not satisfy the continuity equation
Ii)
Differentiating
For two-dimensional flow continuous equation is
Satisfy the continuous equation.
Q16) The velocity components for a three-dimensional incompressible flow is given by: Check whether the flow satisfies continuity?
A16)
Foe three-dimensional equation continuous equation is
The flow is satisfied
Q17) The velocity field is given by check whether the flow satisfies continuity?
A17)
Velocity field
Velocity component
For three-dimensional flow
Steady incompressible fluid flow is possible.
Q18) A stream function is given by check if the flow is rotational and satisfies continuity equation
A18)
By definition
Velocity
1) Irritational flow
2) Continuity equation
The flow is not continuous
Q19) Determine whether the following specified flows are rotational or otherwise. Determine the expression for the velocity potential in case of irrotational flow
A19)
A19) For rotational flow
Flow is rotational.
Q20) Explain Velocity Potential function?
A20)
- The velocity potential is defined as scaler function of space and time such that its negative derivative with respect to any directions gives the fluid velocity in that direction.
- It is denoted by Ø (phi)
- Thus, mathematically the velocity potential is defined as
Ø = f (x, y, z, t)
And Ø = f (x, y, z)
u = - Ø / x
v = - Ø / y
w = - Ø / z
Where u, v and w are the components of velocity in the x, y and z directions respectively.
- The negative sign signifies that Ø decreases with an increase in the values of x, y and z. In other words, it indicates that the flow is always in the direction of decreasing Ø.
Q21) The velocity potential function is given by an expression
A21)
1) Find the velocity components in x and y direction
2) Show that represents a possible case of flow
A21)
Case I) the partial derivation of w.r.t x and y are
The velocity components u and v
Case II) The value of represents a possible case of flow if it satisfies the laplace equation
Represents possible case of flow
Q22) The velocity potential function for a two-dimensional flow is at a point P (4, 5). Determine the velocity.
A22)
Velocity potential
Differential w.r.t x
Hence
By definition of
Velocity at P (4, 5)
Flow is continuous