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# Ch3_4 - Chapter 3 Governing equations 3.1 Instantaneous...

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Chapter 3 Governing equations 3.1 Instantaneous equations (incompressible ﬂuid) Conservation of mass ∂u i ∂x i = 0 (3.1) Navier-Stokes ∂u i ∂t + ( u k u i ) ∂x k = 1 ρ ∂p ∂x i + g i + 1 ρ ∂τ ik ∂x k (3.2) with the shear stress given by: τ ij = 2 µs ij = µ ∂u i ∂x j + ∂u j ∂x i (3.3) with s ij the rate of strain tensor s ij = 1 2 ∂u i ∂x j + ∂u j ∂x i (3.4) The term ∂τ ik /∂x k can also be written as µ 2 u i or sometimes as µ∂ 2 u i /∂x 2 k . We have in- cluded the body force per unit mass g i , e.g. due to buyancy or some other mechanism. Conservation of scalar φ ∂φ ∂t + ( u k φ ) ∂x k = ∂x k D ∂φ ∂x k (3.5) The main quantities appearing in these equations are the velocity u i , the density ρ , the pressure p , g i is the body force, τ ik the viscous stress, D is the molecular diffusion coeﬃcient for φ (e.g. the thermal diffusivity α = λ ρc p if φ is the temperature). Various non-dimensional numbers can be formed from the parameters that appear in the above equations. The main ones are the Prandtl number and the Schmidt number, defined by: 19

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Pr = µc p λ = ν α , ν = µ ρ , α = λ ρc p (3.6) Sc = µ ρD = ν D (3.7) and of course the Reynolds number, based on a suitable velocity and length scale. 3.2 Mean ﬂow equations 3.2.1 Averaging rules Once we have established the main systems of equations governing the instantaneous veloc- ities and scalar concentrations, we can construct equations for the averaged quantities by a straightforward Reynolds averaging procedure. Hence, we write: u i = u i + u i φ = φ + φ p = p + p and we use the averaging rules: φ = 0 φ = φ ∂φ ∂t = φ ∂t and similarly for spatial and higher order derivatives and vector quantities. It is easy to show that these rules are fully consistent with the rules concerning averaging with the PDF (Chapter 2). 3.2.2 First moments Continuity u i ∂x i = 0 (3.8) Subtracting from Eq. (3.1) shows that ∂u i /∂x i = 0 also. Average momentum u i ∂t + u j u i ∂x j = 1 ρ p ∂x i + 2 ν s ij ∂x j ( u i u j ) ∂x j + g i (3.9) 20
In the above, s ij = 1 2 u i ∂x j + u j ∂x i (3.10) is the mean rate of strain and use has been made of continuity to expand the second term in the l.h.s. (the advection). The term 2 ν∂ s ij /∂x j can also be written as ν∂ 2 u i /∂x 2 k or ν 2 u i . Average scalar φ ∂t + u i φ ∂x i = ∂x i D φ ∂x i ( u i φ ) ∂x i (3.11) Reynolds stresses and turbulent ﬂuxes The quantity ρ u i u j is called the Reynolds stress and denotes a ﬂux of i th momentum in the j th direction due to the random velocity in the j th direction. It is called a stress by analogy with the mean viscous stress as it appears in similar form in Eq. (3.9). The quantity u i φ is called turbulent scalar ﬂux and denotes ﬂux of φ in the j th direction due to u j . It is analogous to the molecular ﬂux D∂φ/∂x j .

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Ch3_4 - Chapter 3 Governing equations 3.1 Instantaneous...

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