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Introduction to Computational Fluid Dynamics Course Notes (CFD 4) Karthik Duraisamy Department of Aerospace Engineering University of Glasgow

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Contents Introduction (1.5) Classification of PDE, Model equations (1.5) Finite difference methods: Spatial discretization (3) Temporal discretization (2) Convergence, Consistency, Stability (2) Grids/Boundary conditions (1) RANS Equations and Turbulence modeling (2) DNS/LES (1) Best practices in CFD (1) Case studies/Demonstrations (3) (.) – Approximate number of lectures

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Numerical Simulations : Estimates of cost of fixed wing calculation Sample fixed wing of AR=10, Re=5e6

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Turbulent flow and the NS Equations

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Characteristics of turbulent flow Irregularity (Random and chaotic nature of flow) Increased exchange of momentum (Diffusivity - spreading rate of jets, boundary layers etc.) Large Reynolds numbers Dissipation of kinetic energy to internal energy Wide range of time and length scales Almost all practical flows are turbulent.

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Scales of turbulence Turbulent flows are characterized by a wide range of length scales Think of turbulent flow as a collection of eddies of different sizes Eddies time/length/velocity scales The largest “energy containing” eddies are of the order of the length scale of the object that generated turbulence in the first place (vortex shedding from a cylinder, boundary layer thickness) The smallest eddies are the ones where the energy is dissipated Roughly speaking, there is a “cascade” of energy from the large scale to the smallest scales This happens because the large eddies interact with each other and breakdown into smaller eddies The smallest scales are called the “kolmogorov” scales (η). At these scales, the Reynolds number of the eddies is small enough that viscous effects become dominant and the energy is dissipated.

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Scales

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MOVIES

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Time and length scales Kolmogorov (1941) showed that L/η ~ (Re) 3/4 T/τ ~ (Re) 1/2 The problem with DNS of turbulent flows is that you have to simulate (or resolve) all these scales and in 3 dimensions Therefore, number of points in each direction ~ (Re) 3/4 Therefore, total number of points ~ (Re) 9/4 Therefore, total number of time steps ~ (Re) 1/2 Therefore, total number of operations ~ (Re) 11/4 Remember, on top of this, you have to do hundreds of operations per point

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RANS and LES DNS is obviously not feasible for high Reynolds number flows, therefore, LES and RANS to the rescue LES: Simulate (resolve) the large scales of turbulence and model the effect of the smaller scales (smaller scales are universal) by adding a subgrid viscosity RANS: Model all scales of turbulence Problems: LES still impractical for many flows, RANS is inaccurate in many flows But: One can get very useful and sometimes accurate, affordable engineering solutions with a good knowledge of the flow and the pros and cons of RANS and LES.

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In Summary DNS : All scales of turbulence are resolved. Therefore, smallest grid size is of the order of the Kolmogorov scale η ~ L/(Re) 3/4 LES: Only the “Large” turbulent scales are resolved. The “smaller” scales are modeled RANS : All the turbulent scales are modeled In DNS, you just solve the Navier Stokes Equations In LES, you solve a filtered version of the Navier-Stokes Equations along with another equation to represent the turbulent small scales In RANS, you solve the averaged version of the Navier-Stokes equation along with another equation to represent all the turbulent scales [The extra equations are called turbulence model equations. Typical turbulence models are k-ε, k-ω, Spalart-Allmaras, etc]

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RANS Equations – What are they? Decompose NSE Finally (Substituting & averaging) Where,

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RANS Therefore, the objective in RANS turbulence modeling is to represent the unknown (Reynolds stress) in terms of the knowns (mean flow) Establishing such a relationship is called the closure model Types of closure models: - Algebraic models (Zero equation) [Mixing length model, Baldwin Lomax,Cebecci Smith] - One equation [K-model, Spalart Allmaras] - Two equation [K-e model, k-w model, SST] - Algebraic Reynolds Stress Models (ASBM) - Full Reynolds stress closures

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Eddy viscosity models Therefore, we determine eddy viscosity at each point and add it to the laminar viscosity Closure problem is therefore, to determine eddy viscosity

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Zero equation models L mix is dependant on the problem. Near a wall, it will be a function of distance to the wall and Reynolds number. Models are reasonably accurate in attached flows Very easy to code up Poor correlation in off design conditions No time history effects Fully local.

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2 Equation models 5 free constants Simple, can work well in a variety of flows, history effects, more non-local Reasonable results in many flows, but separation is a problem Very diffusive, poor convergence near walls

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Reynolds Stress Models Solve directly for Reynolds stresses Therefore, in theory more accurate than other methods – Production term is very important. But still have to model some terms (other equations required for epsilon) Very expensive, convergence is poor near the wall

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Discretization Lets now look in little more detail about the mesh spacing required in various types of flows Turbulent flow Wall bounded flow Free-shear flow

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Wall bounded flows In reality, there exist eddies of a wide range of length scales in a boundary layer. In DNS, we simulate (resolve) all these eddies In LES we simulate some of the “significant” eddies and model the rest Problem is that some of the “significant” eddies are still small In RANS we model all the eddies

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Turbulent Boundary layer Knowledge of boundary layer is essential to understand resolution requirements.

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Turbulent Boundary Layer Scaling U* = u + = u / U*, y + = U* y / ν Inner layer Outer layer Inner layer: u + = y + Outer layer u + = 2.5 ln(y + )+5.5 Only for zero pressure gradient boundary layer on a flat plate

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LES Problem with LES is that the near wall streaks are of size L + = 1000, W + =20, H + =30. Therefore to resolve this, we need Δx + = 100 (streamwise), Δy + = 1 (wall normal), Δz + = 5 (spanwise) When you calculate requirements you find out that it is nearly as expensive as DNS!! Note, this is only for wall bounded flows Remember, in LES, you look to simulate those eddies that are significant. It so happens that in near-wall flows, eddies of this size are significant. In free shear flows, the “significant” eddies are not as small – loosely speaking.

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RANS In RANS, you are only concerned about the wall-normal direction. In this direction, you need Δy + = 1 – In the other two directions, grid sizes can be thousands of times larger and hence the savings. Spacing in other two directions is driven by accuracy considerations Remember, in DNS, only numerical error. In LES and RANS, both numerical and turbulence modeling errors

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