Page 24 - Introduction to Computational Fluid Dynamics

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1.2 TRANSPORT EQUATIONS
is invoked. Newton’s second law of motion, combined with Stokes’s stress laws, 12:20 3
yields three momentum equations for velocity in directions x j (j = 1, 2, 3). Similarly,
the first law of thermodynamics in conjunction with Fourier’s law of heat conduction
(q i,cond =−K ∂T /∂x i ) yields the so-called energy equation for the transport of tem-
perature T or enthalpy h. Using tensor notation, we can state these laws as follows:
Conservation of Mass for the Mixture
∂ρ m ∂(ρ m u j )
+ = 0, (1.1)
∂t ∂x j
Equation of Mass Transfer for Species k
∂(ρ m ω k ) ∂(ρ m u j ω k ) ∂ ∂ω k
+ = ρ m D eff + R k , (1.2)
∂t ∂x j ∂x j ∂x j
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 , (1.3)
∂t ∂x j ∂x j ∂x j ∂x i
Energy Equation – Enthalpy Form

∂(ρ m h) ∂(ρ m u j h) ∂ k eff ∂h

+ = + Q , (1.4)
∂t ∂x j ∂x j C pm ∂x j
where enthalpy h = C pm (T − T ref ), and

Energy Equation – Temperature Form

∂(ρ m T ) ∂(ρ m u j T ) ∂ k eff ∂T Q
+ = + . (1.5)
∂t ∂x j ∂x j C pm ∂x j C pm
In these equations, the suffix m refers to the fluid mixture. For a single-
component fluid, the suffix may be dropped and the equation of mass transfer
becomes irrelevant. Similarly, the suffix eff indicates effective values of mass dif-
fusivity D, viscosity µ, and thermal conductivity k. In laminar flows, the values
of these transport properties are taken from property tables for the fluid under
consideration. In turbulent flows, however, the transport properties assume values
much in excess of the values ascribed to the fluid; moreover, the effective transport
properties turn out to be properties of the flow [39], rather than those of the fluid.
From the point of view of further discussion of numerical methods, it is indeed
a happy coincidence that the set of equations [(1.1)–(1.5)] can be cast as a single
equation for a general variable . Thus,
∂(ρ m ) ∂(ρ m u j ) ∂ ∂
+ =
eff + S . (1.6)
∂t ∂x j ∂x j ∂x j