Reynolds - Tony Burdens Lecture Notes on Turbulence Spring...

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Tony Burden’s Lecture Notes on Turbulence, Spring 2007 Chapter 2. Reynolds Equation and the Reynolds Stress In this chapter the basic equations of fluid mechanics are subjected to averaging and in particular Reynolds’ equation for the mean velocity is derived. The calculations are really quite straightforward but the combination of continuum mechanics and statistical analysis tends to lead to lengthy equations. All the flows in these lectures are incompressible. 2a The Basic Equations For the sake of compactness, Cartesian coordinates, x 1 , x 2 , x 3 , and components, u 1 , u 2 , u 3 , will be used in all general analysis in these lecture notes. In specific examples of turbulent flows, the clearer notation x , y , z and u , v , w will be used. x 1 = x u 1 = u x = u U 1 = U x = U x 2 = y u 2 = u y = v U 2 = U y = V x 3 = z u 3 = u z = w U 3 = U z = W The summation convention will be used so that summation signs will not be written explicitly but will be implied by repeated indices; in scalar products of vectors, a · b = 3 i =1 a i b i a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 = a x b x + a y b y + a z b z , and in the divergence operator, ∇ · U = 3 i =1 ∂x i U i ∂x i U i = ∂U ∂x + ∂V ∂y + ∂W ∂z . The continuity condition Conservation of mass requires the mass density, ρ ( x, t ), and the velocity, u ( x, t ), to satisfy, ∂ρ ∂t + ∇ · ( ρ u ) = 0 . In incompressible flow, ρ is constant and conservation of mass leads to the kinematic condition, ∇ · u = ∂u i ∂x i = 0 , which is known as the continuity condition. 7
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The Navier-Stokes equation Newton’s second law, d d t ( mv ) = F , together with the concept of stress in a continuum leads to the equation of motion for a fluid, ∂t ( ρu ) + ∇ · ( ρuu ) = ∇ · τ, or in Cartesian components, ∂t ( ρu i ) + ∂x j ( ρu i u j ) = ∂x j τ ij , where τ is the stress field in the fluid, a second-order tensor. Using conservation of mass, the LHS can be rewritten in the form, ∂t ( ρu i ) + ∂x j ( ρu i u j ) = ρ ∂u i ∂t + ρu j ∂u i ∂x j , so that, ρ ∂u i ∂t + ρu j ∂u i ∂x j = ∂x j τ ij . In the clearer notation x , y , z , u , v , w , the 1- or x - component of this equation is, ρ ∂u ∂t + ρu ∂u ∂x + ρv ∂u ∂y + ρw ∂u ∂z = ∂x τ xx + ∂y τ xy + ∂z τ xz , the x -component of Cauchy’s general equation of motion for a fluid. In inviscid flow, τ ij = - p δ ij , where p is the pressure (and δ ij is equivalent to the 3 × 3 unit matrix). The equation of motion is, ∂u i ∂t + u j ∂u i ∂x j = - 1 ρ ∂p ∂x i , i.e. Euler’s equation. In a Newtonian fluid in incompressible flow, τ ij = - p δ ij + 2 μs ij , where μ is the dynamic viscosity and, s ij = 1 2 ∂u i ∂x j + ∂u j ∂x i , is the rate-of-strain tensor. The equation of motion for a Newtonian fluid is now, ∂t ( ρu i ) + ∂x j ( ρu i u j ) = ∂x j - p δ ij + μ ∂u i ∂x j + ∂u j ∂x i .
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