Report

Numerical Hydraulics Lecture 2: Turbulence Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa Problems of solving the NavierStokes equations • Analytical solutions only known for simple borderline cases (e.g. laminar pipe flow) • Equations non-linear (advective acceleration) • Flow becomes turbulent • Direct numerical solution (DNS) today only possible for relatively low Reynolds numbers • Resolution required for fully developed turbulent flow: roughly 1/1000 of domain size (i.e. in 3D 109 nodes) Turbulent eddy structure turbulent jet turbulent wake past a leaking oil tanker turbulent wake behind a bluff body Turbulent length scales Large scales Small scales • are produced by average flow • depend on boundary conditions and geometry • show structures • inhomogeneous and anisotropic • long-lived and rich in energy • diffusive • difficult to model • no universal model available • are generated by the big scales • are universal • are random • homogeneous and isotropic • short-lived, carrying little energy • dissipative • easy to model • universal model feasible Classification of methods • RANS with stochastic turbulence models (Reynolds averaged Navier-Stokes) – Modelling of the complete turbulent spectrum • LES (Large eddy simulation) – Computation of large scales, modelling of small scales – Compromise between RANS and DNS • DNS (Direct numerical simulation) – Computation of all length scales Classification of methods Degree of modeling 100% RANS LES DNS 0% Computational effort low high extremely high Strong and weak points of DNS – No closure problem of turbulence, no turbulence model necessary – Resolution of all characteristic scales necessary – Problem: Ratio of small turbulence elements (lk) in comparison to large elements (L) grows fast with Renumber: L/lk ~ Re3/4 – Number of grid points NG as well as number of arithmetic operations Nop grow fast with Re: NG ~ Re9/4 , Nop ~ Re11/4 – DNS requires extremely high computer power and storage – DNS in the foreseeable future limited to small Re (Re<104) – DNS still great tool for basic research – DNS suitable to produce reference data for validation of other methods Limitations of DNS • Example: forward facing step (Le & Moin 1992) Extrapolation Grid points Computation time days days years years Reynolds equations (RANS) • Point of departure: Engineers often want to know average properties of flow only • Reynolds decomposition of basic variables into time averaged flow and turbulent fluctuations u u u ' p p p' • Then time-averaging of NS-equations. In all terms which are linear in u and p this is equivalent to replacing the momentary quantity by the time-averaged quantity. • In the non-linear term (advective acceleration) the problem of closure appears….. Reynolds equations additional term ( u ) ( u u ) ( u ' u ') p g u t • The term (u ' u ') requires a closure hypothesis (Turbulence model) • The term (u ' u ') can be interpreted as eddy viscosity • Simplest model: ( u ' u ') eddy u • In contrast to the molecular viscosity, the eddy viscosity is not a material constant, but a function of the flow field itself Turbulence models • Eddy viscosity model – Closure with two equations: e. g. k-e model Further transport equations for k and e are required ' ' k ui'ui' / 2 und and e ui / xk ui / xk i ,k k is turbulent kinetic energy, e is turbulent dissipation rate of energy From both quantities the eddy viscosity is calculated and inserted into the Reynolds equation turb C k2 e empirical C 0.09 empirisch Principle of Large Eddy Simulation (LES) • Decomposition of flow into two parts: Coarse structure (scale L) and fine structure (scale lk) • Coarse structure is computed directly, find structure is modelled Resolvable part Coarse structure (grid scale) - large energy-rich eddies - strongly problem dependent - computed by numerical method Unresolved part Fine structure (subgrid scale) - small eddies with low energy - main effect: energy dissipation - at given resolution nor computable by numerical method - modelling necessary Principle of LES (2) • Starting point: Navier-Stokes equations • Introduction of a filter f ( x, t ) f ( x, t ) f ( x, t ) f = flow quantity (e.g. velocity u) f = resolvable part (coarse structure) f = unresolved part (fine structure) = filter width, G = filter function • Intuitive image: Grid of filter width “fishes” from flow the large, energy rich eddy elements, while the small eddies “escape” through the grid mesh. Large eddies, coarse structure Small eddies, fine structure LES: equations • Filtering of NS-equations (here: incompressible) yields: additional term • Filtering of non-linear terms leads to an additional fine structure stress tensor • Fine structure stress tensors are only formally similar to Reynolds stresses – Reynolds: Decomposition into temporal average and momentary deviation – Filter in LES: Decomposition into coarse structure which can be resolved by numerical method and fine structure which cannot – Fine structure terms vanish for 0 – There is a continuous transition from LES to DNS LES: fine structure models (1) • Tasks of the fine structure model – Modelling the influence of turbulent fine structure on the coarse structure – Modelling the energy transfer between the resolved scales and the unresolved scales in the correct order of magnitude (fine scales have to extract the right amount of energy from the large scales) • Classification of fine structure models – – – – Zero equation models (algebr. eddy viscosity models) One equation models Two equation models Etc. LES: fine structure models (2) One example: Eddy viscosity model by Smagorinski (1963) turb ij ui u j turb x x i j 2 turb CS S mit with 0.16- , 0.2, Filterskala filter scale and CS 0.1 und S 2 (ui / x j )(u j / xi ) i, j Length scale is determined by discretization lengths of grid Main problem of statistical turblence models of determining length scale l does not exist in LES However, Cs is somewhat problem dependent (between 0.065 and 0.2) Summary LES • • • • LES is middle way between DNS and RANS In LES only small scale turbulence has to be modelled Models are simpler and more universal than in RANS LES is particularly suited for complex flows with large scale structures • LES more and more interesting for engineering • Many details still have to be solved (e.g. Boundary conditions) • Growing computer power makes LES more and more attractive Spatially integrated Reynolds equations: Pipe flow • Pipe axis in x-direction a x (pipe axis) • Components of Reynolds-equation in y,zdirections degenerate into pressure eqations • One momentum equation in x-direction remains Spatially integrated Reynolds equations: Pipe flow 1 Q um ux dA AA A • x-component of Reynolds equation ux 1 ux p ux g sin a f x dA 0 A A t x x • Transverse components and inner friction cancel out during integration. What remains is the wall shear stress Spatially integrated Reynolds equations: Pipe flow • Required: Continuity equation for elastic pipes, expression for wall shear stress (Turbulence model) • Wall shear stress (see Hydraulics I): um2 x r 2 x um2 h pA 0 2 r x 2r 2 g 4r 0 2 force 0 um and 8 volume Rhy Spatially integrated Reynolds equations: Pipe flow • The cross-sectionally averaged momentum equation in x-direction thus develops into um um pm 0 a ' um g sin a 0 t x x Rhy 1 1 2 with a ' 2 um dA um A A Spatially integrated Reynolds equations: Pipe flow • Or with the friction slope IR and a‘=1: IR 0 gRhy um um pm um g sin a gI R 0 t x x Spatially integrated Reynolds equations: Pipe flow • Continuity equation for an elastic pipe: – In the mass balance per unit length of pipe, the cross sectional area A is not a constant • Therefore: ( A) ( Aum ) 0 t x um A um A um 0 t A t x A x x with ( p) and A A( p) Spatially integrated Reynolds equations: Open channel flow • Additional assumptions: hydrostatic pressure distribution p=g(hp - z), cosa ≈ 1, sina = -dz/dx, = constant lead to: 0 ( pm / g ) z 1 um uma ' um 0 g t g x g Rhy x x um um h a ' um gI R g gI S 0 t x x Spatially integrated Reynolds equations: Open channel flow • The cross-sectional area is again a function of time. • The continuity equation is written as: A Q 0 t x