Page 299 - Introduction to Computational Fluid Dynamics

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APPENDIX A. DERIVATION OF TRANSPORT EQUATIONS
respectively. Such equations are called nonconservative forms of momentum
equations.
2. Equations A.5, A.11, A.12, and A.13 define the fluid motion completely.
However, they contain twelve unknowns (three velocity components and nine
stresses). By invoking the rule of complementarity of stresses (i.e., τ ij = τ ji , i =
j), the unknowns can be reduced to nine. Still, the number of unknowns exceeds
the number of available equations (four).
3. A solvable system must have the same number of unknowns and equations. To
do this, Stokes’s stress laws are invoked:
Stress Laws

∂u i ∂u j
τ ij = µ + , (A.14)
∂x j ∂x i

∂u i

σ i =−p + σ =−p + 2µ (no summation), (A.15)
i
∂x i
2

where σ is called the deviatoric stress, p is pressure (compressive), and µ is
i
the viscosity of the fluid. 3
4. When Equations A.14 and A.15 are substituted in Equations A.11, A.12, and
A.13, the new equations can be compactly written in tensor notation as
Momentum Equations u i (i =1,2,3)
∂(ρ m u i ) ∂(ρ m u j u i ) ∂ ∂u i ∂p
+ = µ eff − + ρ m B i + S u i . (A.16)
∂t ∂x j ∂x j ∂x j ∂x i
This equation is the same as Equation 1.3 in Chapter 1. The three equations
(A.16) now contain only four unknowns (u 1 , u 2 , u 3 , and p). Along with Equa-
tion A.5, therefore, there are as many unknowns as there are equations.

2 In Chapter 1, the deviatoric stress is expressed as

∂u i
σ = 2 µ + q

i
∂x i
and significance of q is explained in Section 1.5.
3 In turbulent flows, the total stress comprises additive contributions of laminar and turbulent com-
t
ponents. The turbulent stress τ =−ρ m u u is again represented in the manner of Equation A.14

ij i j
by invoking turbulent viscosity µ t . This is known as the Boussinesq approximation. Then the total
stress τ tot in a turbulent flow is given by
ij
2
t
τ ij tot = τ ij + τ = (µ + µ t ) ∂u i + ∂u j − ρ m e δ ij ,
ij
∂x j ∂x i 3
3

where δ ij is the Kronecker delta and e = i=1 u u /2 is the kinetic energy of velocity fluctuations.
i
i