Potential Flow and Potential Flow and Computational Fluid DynamicsComputational Fluid Dynamics

Numerical AnalysisNumerical AnalysisC8.3C8.3

Saleh David RamezaniSaleh David Ramezani

BIEN 301BIEN 301

February 14, 2007February 14, 2007

ProblemProblem GivenGiven: : Consider plane inviscid flow Consider plane inviscid flow

through a symmetric diffuser as shown through a symmetric diffuser as shown below. Only the upper half is shown. below. Only the upper half is shown. The flow is to expand from inlet half-The flow is to expand from inlet half-width h to exit half width 2h. The width h to exit half width 2h. The expansion angle θ is 18.5˚. expansion angle θ is 18.5˚.

ProblemProblem AskedAsked: Set up a non-square : Set up a non-square

potential flow mesh for this problem, potential flow mesh for this problem, and calculate the plot (a) the velocity and calculate the plot (a) the velocity distribution and (b) the pressure distribution and (b) the pressure coefficient along the centerline. coefficient along the centerline. Assume uniform inlet and exit flows. Assume uniform inlet and exit flows.

AssumptionsAssumptions Incompressible flowIncompressible flow Frictionless flowFrictionless flow Neglected gravityNeglected gravity Steady flowSteady flow

Free Body DiagramFree Body Diagram

θ

2h

L2h

hV

hLLh

Lh

3

335.0)5.18tan(

tan

Mesh ModelMesh ModelWe can make our mesh model with We can make our mesh model with

longlongand and high rectangles. high rectangles.

5h

2h

h31

h91

31

x

91

y

Stream FunctionStream Function For a non-square mesh use equation For a non-square mesh use equation

8.108 to find the stream function.8.108 to find the stream function.

wherewhere

)()1(2 1,1,,1,1, jijijijiji

92

yx

Stream FunctionStream Function

Simplify to get:Simplify to get:

)(9201

1,1,,1,1, jijijijiji

)(9)91(2 1,1,,1,1, jijijijiji

Boundary ConditionsBoundary Conditions Boundary values are not given.Boundary values are not given. Assume your own stream boundary Assume your own stream boundary

values along the walls. values along the walls. Choose 100 mChoose 100 m22/s along the top wall, /s along the top wall,

and 0 along the lower wall.and 0 along the lower wall. Use Excel to iterate for stream Use Excel to iterate for stream

function nodal values. function nodal values.

Stream Function Nodal Stream Function Nodal ValuesValues

VelocityVelocity Velocity at any point in the flow can be Velocity at any point in the flow can be

computed from equation 8.107:computed from equation 8.107:

Using Excel and our previously Using Excel and our previously computed Stream nodal values we can computed Stream nodal values we can find the velocity nodal values.find the velocity nodal values.

)(1,1, jijiyy

V

Velocity Nodal ValuesVelocity Nodal Values

Pressure CoefficientPressure Coefficient Pressure coefficient can be computed Pressure coefficient can be computed

from Bernoulli’s equation. from Bernoulli’s equation.

The simplified form of this equation is The simplified form of this equation is found in example 8.5 of the textbook.found in example 8.5 of the textbook.

211

2

21

21 VpVp

Pressure CoefficientPressure Coefficient

V was previously computed from stream V was previously computed from stream function.function.

VV11 is the velocity at the entrances found is the velocity at the entrances found from the stream function (here 200 mfrom the stream function (here 200 m22/s)/s)

2

121

1 1

21

VV

V

ppC p

Pressure Coefficient Nodal Pressure Coefficient Nodal ValuesValues

Stream DistributionStream DistributionStream Distribution

0.05.010.015.020.025.030.035.040.045.050.0

0 2 4 6 8 10 12 14 16

Horizontal Position

Stre

am (m

^2/s

)

Velocity DistributionVelocity DistributionVelocity Distribution

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16

Horizontal Position

Velo

city

(m/s

)

Pressure Coefficient Pressure Coefficient DistributionDistribution

Pressure Coefficient Distribution

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0 2 4 6 8 10 12 14 16

Horizontal Position

Pres

sure

Coe

ffici

ent

Biomedical ApplicationBiomedical Application Not all blood vessels or passage ways in Not all blood vessels or passage ways in

our body have a uniform thickness or our body have a uniform thickness or shape. In fact most of them are shape. In fact most of them are characterized by complicated characterized by complicated geometries. The most obvious geometries. The most obvious biomedical application of this problem is biomedical application of this problem is the numerical analysis of velocities or the numerical analysis of velocities or flow rates through those more complex flow rates through those more complex shaped passage ways. shaped passage ways.