Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

150

The viscosity stress tensor is determined by the expressions:

2 3 µ ( u x + v y + w z ) , 2 3 µ ( u x + v y + w z ) , 2 3 µ ( u x + v y + w z ) ,

τ xx = 2 µ u x − τ yy = 2 µ v y − τ zz = 2 µ w z −

τ xy = τ yx = µ ( u y + v x ) , τ xz = τ zx = µ ( u z + w x ) , τ yz = τ zy = µ ( v z + w y ) ,

(4.127.1)

while the vector of the thermal flux is:

∂ T ∂ y

∂ T ∂ z

∂ T ∂ x

Q x = − k

Q y = − k

i Q z = − k

(4.127.2)

,

,

where the viscosity coefficient µ is a function of temperature T

T T ∞

) 0 . 67 .

µ = µ ∞ (

(4.127.3)

The coefficient of thermal conductivity k is defined by the relation

γ γ − 1

µ Pr

k =

(4.127.4)

.

Prandtl number Pr in the expression (4.127.4) can be considered constant, ie. value Pr = 0 . 72 is accepted, while γ = 1 . 4 is the value of the air adiabatic constant. In the case of turbulent flow the value of the turbulent coefficient µ t is added to the laminar viscosity coefficient µ while in the expression for the thermal conductivity coefficient k the ratio µ / Pr is replaced by sum µ / Pr + µ t / Pr t , where Pr t = 0 . 9 is Prandtl number for turbulent flow. Coefficient of turbulent viscosity µ t is calculated on the basis of generally accepted Baldwin-Lomax model [4], [2]. After the introduction of the curvilinear coordinate system ( ξ , η , ζ ) , the generated grid in physical space is mapped to a rectangle computational grid. If only partial derivatives of flow quantities in viscous terms in the direction normal to body surface are retained, the approximate Navier-Stokes equations are reached, ie. thin layer equation are obtained. For that matter, equation (4.126) in the transformed space becomes ∂ τ q + ∂ ξ F + ∂ η G + ∂ ζ H = ∂ ζ H v , (4.126.1) where the quantities q , F , G and H are defined by relations (4.14.1) and (4.15.2), Viscous term on the right hand of equation (4.126.1) after coordinate transformation takes the form

   

    ,

0

µ ( ζ 2 µ ( ζ 2 µ ( ζ 2 2 z ) u ζ + µ / 3 ( ζ x u ζ + ζ y v ζ + ζ z w ζ ) ζ x 2 z ) v ζ + µ / 3 ( ζ x u ζ + ζ y v ζ + ζ z w ζ ) ζ y 2 z ) w ζ + µ / 3 ( ζ x u ζ + ζ y v ζ + ζ z w ζ ) ζ z µ / 2 ( ζ 2 2 + v 2 + w 2 ) ζ + + µ / 3 ( ζ x u + ζ y v + ζ z w ) ( ζ x u ζ + ζ y v ζ + ζ z w ζ )+ + k ( ζ 2 x + ζ 2 y + ζ 2 z ) T ζ x + ζ 2 y + ζ 2 z ) ( u x + ζ x + ζ x + ζ 2 y + ζ 2 y + ζ 2 y + ζ

H v = J

(4.127.5)