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AMME5202 Lecture 9 Summary: Turbulence Turbulence occurs when hydrodynamic instability prevents a steady flow from occurring. Energy which is contained in large scale motions with a size L is passed down to smaller length scales via an energy cascade until the scales are small enough for viscosity to dissipate their energy. One critical feature of this energy cascade is that it relies on three dimensional motion. Two dimensional simulations cannot correctly capture the turbulent energy cascade. The amount of turbulence in a flow is related to the Reynolds number, Re = uL ν . Different situation call for various choices of velocity and length scale used in this cal- culation. Turbulence presents a significant problem for computational fluid dynamics. In order to simulate a turbulent flow, the simulation must use a sufficient grid resolution in order to capture all turbulent length scales of the flow. It is also impossible to use a two dimensional simulation so even flows which appear two dimensional such as the ducts, flows over steps and over airfoils that we have studied must be run using a third dimension. Also the simulation will never reach a steady state, generally will need to run until a large number of time samples are calculated in order to capture time averaged quantities of interest. The smallest length scales are given by the Kolmogorov length scale l K . The ratio between the Kolmogorov length scale and the energy containing scales is, L l K ∼ Re 3 4 In order to illustrate the computational requirements for direct numerical simulation of turbulent flows, let us consider a duct with a typical engineering scale Reynolds number of 100,000. Assuming that we have a grid size equal to the Kolmogorov length scale and a domain size at least equal to the energy containing scale then the number of grid points is roughly proportional to Re 3 4 . Now since we need that many cells in three dimensions, the total number of cells is, n ∼ Re 9 4 = 1 . 8 × 10 11 . If each cell requires the storage of four variables using 8 bytes of computer memory each then that is roughly 5 . 8 × 10 12 bytes or 5.8 TB of memory required. This scale of simulation is still not feasible on the worlds largest supercomputers. This is why turbulence modelling is an important area of research. 1

Tensor notation In the remainder of this lecture we will introduce tensor notation. Tensors are a gen- eralized way to represent scalars, vectors and matrices as well as higher rank tensors which have no equivalent. A scalar is a rank zero tensor, a vector is a rank one tensor and a matrix is a rank two tensor. In this course we will use Greek subscripts to denote tensors, for example a vector is written as, u ˜ = u α , where α can take three values, 1, 2 and 3 corresponding to the x , y , and z directions. In tensor notation summation is implied over repeated indices, for example we would write the dot product as, u ˜ · v ˜ = u α v α = u x v x + u y v y + u z v z . The del operator ∇ ˜ , a vector of derivatives, is written as, ∇ ˜ = ∂ α . The gradient of a scalar is written using the del operator, ∇ ˜ φ = ∂ α φ. The divergence of a vector is written as, ∇ ˜ · u ˜ = ∂ α u α . A matrix vector product can be written as, A αβ u α =   A xx u x + A yx u y + A zx u z A xy u x + A yy u y + A zy u z A xz u x + A yz u y + A zz u z   . A commonly used tensor is the Kronecker delta, δ αβ = 1 : α = β 0 : otherwise The Kronecker delta is equivalent to the identity matrix. Multiplication by the Kronecker delta where one index is repeated is equivalent to replacing the repeated index by the other one, A αβ δ βγ = A αγ The trace of a matrix is a scalar, it may be written as, A αβ δ αβ = A xx + A yy + A zz 2

The cross product makes use of the Levi-Civita symbol of rank and dimension three. It’s value is 0 if any indices are repeated, 1 if the indices are all unique and form a cyclic permutation of { 1 , 2 , 3 } and - 1 if the indices are unique and form an anti-cyclic permutation of { 1 , 2 , 3 } , ε αβγ =      +1 for { α, β, γ } = { 1 , 2 , 3 } or { 2 , 3 , 1 } or { 3 , 1 , 2 } - 1 for { α, β, γ } = { 3 , 2 , 1 } or { 2 , 1 , 3 } or { 1 , 3 , 2 } 0 otherwise The definition of a cross product is, u ˜ × v ˜ = ε αβγ u β v γ . Notice that two of the indices are repeated and hence summed over, one index is not repeated so the result is a vector. The convective form of the Navier-Stokes equations can be written as, ∂ α ρu α = 0 ∂ t u α + u β ∂ β u α = - 1 ρ ∂ α P + ν∂ β ∂ β u α Reynolds-averaged Navier-Stokes equations In order to model the effects of turbulence, the flow variables u α and P are split into mean and fluctuating quantities, ¯ u α = 1 t 1 - t 0 Z t 1 t 0 u α d t Now the fluctuating component is, u 0 α = u α - ¯ u α . The same definitions are used for mean and fluctuating pressure. These definitions are substituted into the Navier-Stokes equations, ∂ α u α = 0 ∂ t u α + ∂ β u α u β = - 1 ρ ∂ α P + ν∂ β ∂ β u α and the time average of the entire set of equations is taken resulting in, ∂ α (¯ u α + u 0 α ) = 0 ∂ t (¯ u α + u 0 α ) + ∂ β ( ¯ u α + u 0 α )( ¯ u β + u 0 β ) = - 1 ρ ∂ α ( ¯ P + P 0 ) + ν∂ β ∂ β (¯ u α + u 0 α ) . 3

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