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:

Подпись:Подпись: (8.53)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.

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