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Compressible Viscous Fluid Flow
5.2.1 Viscous Stresses and Constitutive Relations
For compressible flows, there is a second viscosity coefficient A. Stresses are related to strain rates as follows:
|
д и |
ди |
Dv |
|
|
&1,1 = 2p + A Dx |
Dx + д у |
(8.48) |
|
|
Dv |
ди |
Dv |
|
|
V2,2 = 2p + A д у |
Dx + д у |
(8.49) |
|
|
/ |
‘ди |
Dv |
(8.50) |
|
&1,2 = &2,1 = U |
чд x + д у |
According to Stokes hypothesis: A = — 2p.
5.2.2 Navier-Stokes Equations for 2-D Compressible Flows
Assuming A and u are constant Conservation of mass:
дри Dpv Dx + ду 0
Conservation of x-momentum:
du du dp (д2н d2u p (d2u d2v
pudX + pvdy = ~3x + p oX2 + + 3 dX2 + dXdy
Conservation of y-momentum:
dv dv dp (d2 v d2v p ( d2u d2v
pudX + pvdy = ~d + p oX2 + dy2 + 3 дХдУ + of2
The system is completed with the equation of energy for temperature T and the equation of state (for a perfect gas) p = pRT.
