1. Pharos University MECH 253 FLUID MECHANICS II
Lecture # 5
INVISCID FLOWS

2. 1 Inviscid Flow
Inviscid flow implies that the viscous effect is negligible. This occurs in the flow domain away from a solid boundary outside the boundary layer at Re.
The flows are governed by Euler Equations where , v, and p can be functions of r and t .

3. 7.1 Inviscid Flow
On the other hand, if flows are steady but compressible, the governing equation becomes where  can be a function of r
For compressible flows, the state equation is needed; then, we will require the equation for temperature T also.

4. 7.1 Inviscid Flow
Compressible inviscid flows usually belong to the scope of aerodynamics of high speed flight of aircraft. Here we consider only incompressible inviscid flows.
For incompressible flow, the governing equations reduce to where  = constant.

5. 7.1 Inviscid Flow
For steady incompressible flow, the governing eqt reduce further to where  = constant.
The equation of motion can be rewrited into
Take the scalar products with dr and integrate from a reference at  along an arbitrary streamline =C , leads to since

6. 7.1 Inviscid Flow
If the constant (total energy per unit mass) is the same for all streamlines, the path of the integral can be arbitrary, and in the flow domain except inside boundary layers.
Finally, the governing equations for inviscid, irrotational steady flow are
Since is the vorticity , flows with are called irrotational flows.

7. 7.1 Inviscid Flow
Note that the velocity and pressure fields are decoupled. Hence, we can solve the velocity field from the continuity and vorticity equations. Then the pressure field is determined by Bernoulli equation.
A velocity potential  exists for irrotational flow, such that, and irrotationality is automatically satisfied.

8. 7.1 Inviscid Flow
The continuity equation becomes which is also known as the Laplace equation.
Every potential satisfy this equation. Flows with the existence of potential functions satisfying the Laplace equation are called potential flow.

9. 7.1 Inviscid Flow
The linearity of the governing equation for the flow fields implies that different potential flows can be superposed.
If 1 and 2 are two potential flows, the sum =(1+2) also constitutes a potential flow. We have
However, the pressure cannot be superposed due to the nonlinearity in the Bernoulli equation, i.e.

10. Potential Flows Integral Equations
Irrotational Flow
Flow Potential
Conservation of Mass
Laplace Equation

11. 7.2 2D Potential Flows
If restricted to steady two dimensional potential flow, then the governing equations become
E.g. potential flow past a circular cylinder with D/L <<1 is a 2D potential flow near the middle of the cylinder, where both w component and /z0. U L y x z D

12. 7.2 2D Potential Flows
The 2-D velocity potential function gives and then the continuity equation becomes
The pressure distribution can be determined by the Bernoulli equation, where p is the dynamic pressure

13. 7.2 2D Potential Flows
For 2D potential flows, a stream function (x,y) can also be defined together with (x,y). In Cartisian coordinates, where continuity equation is automatically satisfied, and irrotationality leads to the Laplace equation,
Both Laplace equations are satisfied for a 2D potential flow

14. 7.2 Two-Dimensional Potential Flows
For two-dimensional flows, become:
In a Cartesian coordinate system
In a Cylindrical coordinate system
and
and

15. Taking into account:
Continuity equation

16. Irrotational Flow Approximation
For 2D flows, we can also use the stream function
Recall the definition of stream function for planar (x-y) flows
Since vorticity is zero,
This proves that the Laplace equation holds for the stream function and the velocity potential

17. Cylindrical coordinate system
In cylindrical coordinates (r , q ,z ) with
-axisymmetric case

18. Taking into account:
Continuity equation

19. 7.2 Two-Dimensional Potential Flows
Therefore, there exists a stream function such that
in the Cartesian coordinate system and
in the cylindrical coordinate system.
The transformation between the two coordinate systems

20. 7.2 Two-Dimensional Potential Flows
The potential function and the stream function are conjugate pair of an analytical function in complex variable analysis. The conditions:
These are the Cauchy-Riemann conditions. The analytical property implies that the constant potential line and the constant streamline are orthogonal, i.e.,
and to imply that

21. Irrotational Flow Approximation
Irrotational approximation: vorticity is negligibly small
In general, inviscid regions are also irrotational, but there are situations where inviscid flow are rotational, e.g., solid body rotation

22. 7.3 Simple 2-D Potential Flows
Uniform Flow
Stagnation Flow
Source (Sink)
Free Vortex

23. 7.3.1 Uniform Flow For a uniform flow given by , we have Therefore,
Where the arbitrary integration constants are taken to be zero at the origin.
and
and

24. 7.3.1 Uniform Flow
This is a simple uniform flow along a single direction.

25. Elementary Planar Irrotational Flows Uniform Stream
In Cartesian coordinates
Conversion to cylindrical coordinates can be achieved using the transformation
Proof with Mathematica

26. 7.3.2 Stagnation Flow
For a stagnation flow, Hence,
Therefore,

27. 7.3.2 Stagnation Flow
The flow an incoming far field flow which is perpendicular to the wall, and then turn its direction near the wall
The origin is the stagnation point of the flow. The velocity is zero there. x  y

28. 7.3.3 Source (Sink)
Consider a line source at the origin along the z-direction. The fluid flows radially outward from (or inward toward) the origin. If m denotes the flowrate per unit length, we have (source if m is positive and sink if negative).
Therefore,

29. 7.3.3 Source (Sink) The integration leads to and
Where again the arbitrary integration constants are taken to be zero at
and

30. 7.3.3 Source (Sink)
A pure radial flow either away from source or into a sink
A +ve m indicates a source, and –ve m indicates a sink
The magnitude of the flow decrease as 1/r
z direction = into the paper. (change graphics)

31. Elementary Planar Irrotational Flows Line Source/Sink
Potential and streamfunction are derived by observing that volume flow rate across any circle is
This gives velocity components

32. Elementary Planar Irrotational Flows Line Source/Sink
Using definition of (Ur, U)
These can be integrated to give  and 
Equations are for a source/sink
at the origin
Proof with Mathematica

33. 7.3.4 Free Vortex
Consider the flow circulating around the origin with a constant circulation . We have: where fluid moves counter clockwise if is positive and clockwise if negative.
Therefore,

34. 7.3.4 Free Vortex
The integration leads to
where again the arbitrary integration constants are taken to be zero at
and

35. 7.3.4 Free Vortex
The potential represents a flow swirling around origin with a constant circulation .
The magnitude of the flow decrease as 1/r.

36. 7.4. Superposition of 2-D Potential Flows
Because the potential and stream functions satisfy the linear Laplace equation, the superposition of two potential flow is also a potential flow.
From this, it is possible to construct potential flows of more complex geometry.
Source and Sink
Doublet
Source in Uniform Stream
2-D Rankine Ovals
Flows Around a Circular Cylinder

37. 7.4.1 Source and Sink
Consider a source of m at (-a, 0) and a sink of m at (a, 0)
For a point P with polar coordinate of (r, ). If the polar coordinate from (-a,0) to P is and from (a, 0) to P is
Then the stream function and potential function obtained by superposition are given by:

38. 7.4.1 Source and Sink

39. 7.4.1 Source and Sink
Hence,
Since
We have

40. 7.4.1 Source and Sink
We have By
Therefore,

41. 7.4.1 Source and Sink
The velocity component are:

42. Elementary Planar Irrotational Flows Doublet
A doublet is a combination of a line sink and source of equal magnitude
Source
Sink

43. 7.4.1 Source and Sink

44. 7.4.2 Doublet
The doublet occurs when a source and a sink of the same strength are collocated the same location, say at the origin.
This can be obtained by placing a source at (-a,0) and a sink of equal strength at (a,0) and then letting a  0, and m , with ma keeping constant, say 2am=M

45. 7.4.2 Doublet For source of m at (-a,0) and sink of m at (a,0)
Under these limiting conditions of a0, m , we have

46. 7.4.2 Doublet Therefore, as a0 and m with 2am=M
The corresponding velocity components are

47. 7.4.2 Doublet

48. 7.4.3 Source in Uniform Stream
Assuming the uniform flow U is in x-direction and the source of m at(0,0), the velocity potential and stream function of the superposed potential flow become:

49. 7.4.3 Source in Uniform Stream

50. 7.4.3 Source in Uniform Stream
The velocity components are:
A stagnation point occurs at
Therefore, the streamline passing through the
stagnation point when
The maximum height of the curve is

51. 7.4.3 Source in Uniform Stream
For underground flows in an aquifer of constant thickness, the flow through porous media are potential flows.
An injection well at the origin than act as a point source and the underground flow can be regarded as a uniform flow.

52. D Rankine Ovals
The 2D Rankine ovals are the results of the superposition of equal strength sink and source at x=a and –a with a uniform flow in x-direction.
Hence,

53. D Rankine Ovals
Equivalently,

54. 7.4.4 2-D Rankine Ovals The stagnation points occur at
where with corresponding

55. 7.4.4 2-D Rankine Ovals The maximum height of the Rankine oval is
located at when ,i.e.,
which can only be solved numerically.

56. D Rankine Ovals rs ro

57. 7.4.5 Flows Around a Circular Cylinder
Steady Cylinder
Rotating Cylinder
Lift Force

58. Steady Cylinder
Flow around a steady circular cylinder is the limiting case of a Rankine oval when a0.
This becomes the superposition of a uniform parallel flow with a doublet in x-direction.
Under this limit and with M=2a. m=constant,
is the radius of the cylinder.

59. Steady Cylinder
The stream function and velocity potential become:
The radial and circumferential velocities are:

60. Steady Cylinder ro

61. Examples of Irrotational Flows Formed by Superposition
Flow over a circular cylinder: Free stream + doublet
Assume body is  = 0 (r = a)  K = Va2

62. Examples of Irrotational Flows Formed by Superposition
Velocity field can be found by differentiating streamfunction
On the cylinder surface (r=a)
Normal velocity (Ur) is zero, Tangential velocity (U) is non-zero slip condition.

63. Rotating Cylinder
The potential flows for a rotating cylinder is the free vortex flow. Therefore, the potential flow of a uniform parallel flow past a rotating cylinder at high Reynolds number is the superposition of a uniform parallel flow, a doublet and free vortex.
Hence, the stream function and the velocity potential are given by

64. Rotating Cylinder
The radial and circumferential velocities are given by

65. Rotating Cylinder
The stagnation points occur at
From

66. Rotating Cylinder

67. Rotating Cylinder

68. Rotating Cylinder The stagnation points occur at Case 1: Case 2:

69. Rotating Cylinder
Case 1:

70. Rotating Cylinder Case 2:
The two stagnation points merge to one at cylinder surface where

71. Rotating Cylinder Case 3:
The stagnation point occurs outside the cylinder
when where The condition of
leads to
Therefore, as , we have

72. Rotating Cylinder
Case 3:

73. Lift Force
The force per unit length of cylinder due to pressure on the cylinder surface can be obtained by integrating the surface pressure around the cylinder.
The tangential velocity along the cylinder surface is obtained by letting r=ro,

74. Lift Force The surface pressure as obtained from Bernoulli equation is
where is the pressure at far away from the cylinder.

75. Lift Force
Hence,
The force due to pressure in x and y directions are then obtained by

76. Lift Force
The lift on rotating bodies is called the Magnus effect.
the lift force is due to the circulation around the body.
An airfoil without rotation can develop a circulation around the airfoil when Kutta condition is satisfied at the rear tip of the airfoil.
Therefore, The tangential velocity along the cylinder surface is obtained by letting r=ro:
This forms the base of aerodynamic theory of airplane.