Direct numerical simulation. in an annular pipe. of turbulent flow. Paolo Luchini & Maurizio Quadrio
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1 P.Luchini & M.Quadrio SIMAI Ischia Direct numerical simulation of turbulent flow in an annular pipe Paolo Luchini & Maurizio Quadrio Dipartimento di Ingegneria Aerospaziale del Politecnico di Milano via La Masa Milano maurizio.quadrio@polimi.it
2 P.Luchini & M.Quadrio SIMAI Ischia Why the annular pipe? Incomplete experimental information (additional measuring difficulties in near-wall region) regarding turbulence statistics Effect of transverse curvature on turbulence not fully documented Numerical solution of incompressible NS equation (DNS) Objectives Extend to the cylindrical geometry a numerical method for the DNS of turbulent plane channel flow (adapt a computer code without structural changes) Perform DNS of the turbulent flow in an annular pipe (never reported)
3 Very demanding in terms of computational resources, even at low/moderate modeling) Research tool, suited for basic turbulence research (flow physics, turbulence DNS of practical flows at high is unaffordable Requirements of spatio-temporal resolution rise with for Full range of spatial & temporal scales in turbulent flows needs to be accounted Numerical solution of the full Navier Stokes equations (vs. LES or RANS) What is DNS of turbulence? P.Luchini & M.Quadrio SIMAI Ischia
4 P.Luchini & M.Quadrio SIMAI Ischia DNS of turbulence in cylindrical coordinates Numerical difficulties of cylindrical coordinate system few DNS of turbulent flow in cylindrical geometries (pipe flow only) Many of the numerical schemes solve NS eqs in primitive variables, with the pressure-correction approach Only one DNS study (Neves, Moin & Moser, JFM v.272, 994) considers transverse curvature, but is concerned with the boundary layer over small cylinders (with insufficient resolution in the outer part of the layer) The comparison between the effects of convex / concave transversal curvature on the turbulence statistics is still missing
5 P.Luchini & M.Quadrio SIMAI Ischia The (standard) cartesian case For plane channel flow, there is an almost standard procedure developed by Kim, Moin & Moser (JFM v.77, 97), by which: Pressure is eliminated from the equations NS system is reduced to a second-order scalar equation for the normal vorticity and a fourth-order scalar equation for the normal velocity When using Fourier transforms in homogeneous directions, the other velocity components are easily recovered High (nearly optimal) computational efficiency can be achieved
6 P.Luchini & M.Quadrio SIMAI Ischia
7 *!!! " " $ % %'& ( )! $ % %'& ( ) *! $ % % & ( ) Initial conditions. Periodicity in and. No-slip boundary conditions at the walls: # By manipulating momentum equation, and using continuity, one obtains for : component, one obtains for : By applying to the momentum equation,. Considering the Cartesian case (cont.) P.Luchini & M.Quadrio SIMAI Ischia
8 /! ! 2 - -, 0 smaller than those needing accurate representation) convective part, but implicit schemes for the viscous terms (with time-scales Partially implicit approach is very popular: explicit schemes for the wide range of spatial scales Implicit time schemes are usually not used, due to the need of accuracy over a 2 2 recovered with the solution of a 2x2 algebraic system: By Fourier transforming in the homogeneous directions, and are easily Cartesian case (cont.) P.Luchini & M.Quadrio SIMAI Ischia
9 where $ ) Cylindrical case P.Luchini & M.Quadrio SIMAI Ischia
10 P.Luchini & M.Quadrio SIMAI Ischia Cylindrical case (cont.) flow L θ R e r, v L x R 2δ i x, u θ, w
11 9 * ; " : 9 7 : Notation: 7 6 depends on! equation are coupled impossible implicit treatment of viscous terms: components of the momentum ) and4(wave number 6) these difficulties are left: By Fourier transforming the equations in homogeneous directions (wave number Cylindrical case (cont.) P.Luchini & M.Quadrio SIMAI Ischia
12 ? > 0 > > > < = presumably small since difference with cartesian, hence no stability problems Both and terms can enter the explicit part: low-order derivatives, Contains a curvature term Not independent of anymore (no problem if equation for does not contain!) > 6 considering the component, one obtains for radial vorticity : In analogy with the cartesian case, by taking of the momentum equation and The method: equation for P.Luchini & M.Quadrio SIMAI Ischia
13 A B A B : B " 7 7 : 7 DC DC 7 : : 6 A 7 : Solve for, then put incomponent of momentum eq. Further simplifications by using continuity Expressions for and are taken from momentum eq. Continuity equation is Fourier transformed and time differenced The method: equation for P.Luchini & M.Quadrio SIMAI Ischia
14 / > 60 > - -, 0 60? > equation are to be solved directly. As in cartesian case, for 7 the and4components of the momentum 7 : 7 : algebraic system: In analogy with the cartesian case, e are recovered with the solution of a 2x2 Curvature terms can enter the explicit part, without stability problems Contains curvature terms Independent of The method: equation P.Luchini & M.Quadrio SIMAI Ischia
15 P.Luchini & M.Quadrio SIMAI Ischia The numerical solution FFT algorithms allow exact computation of the nonlinear terms in physical space; De-aliasing with the /2 rule. Radial derivatives discretized with finite differences over a -point stencil; low-order derivatives are IV order, higher order derivatives are IInd order formally IInd order method, but advantageous compared to a IInd order scheme Time integration: as in KMM, third-order Runge-Kutta for the explicit part, and second-order Crank-Nicholson for the implicit part.
16 V V Q R ( Q, where R 6 I 6 I which gives I L M E F E G H E G E F K Q R " R " (, storing flow fields every K Q ( Constant flow rate; after reaching steady state, computations are carried out for value for channel flow) for the inner wall, and XWfor the outer wall ( U reference "TS U is bulk velocity; corresponds to V? U H spanwise dimensions wider than plane case No need to consider full circumferential extension of the annulus: gives N ( O L P O H N ((inner / outer walls) J Kwhich gives H N ( Inner radius (and outer radius (; gap ( Periodicity assumption in and4directions The physical problem P.Luchini & M.Quadrio SIMAI Ischia
17 6 I P.Luchini & M.Quadrio SIMAI Ischia Transversal resolution & cylindrical coordinates For a given, transversal size of the computational domain increases with Physical considerations dictate minimal required resolution, needed at outer wall Resolution increases (linearly) above necessary approaching inner wall ;potential stability problems, and waste of computational resources With Fourier schemes, Fourier series must be truncated at wavenumber corresponding to maximum resolution Solution An -dependent truncation of the series can remove the unneeded azimuthal modes, saving memory and CPU time, and avoiding stability problems.
18 Q? [? [ [ E F E G O " XY R " Time step J ((comparable to planar case); computing time? weeks parallel speedup of 00% 20 seconds / time step for a SMP personal computer (2 CPU Intel 0MHz); 2 millions d.o.f.; RAM memory: 0MB; single flow field on disk: 0MB Spatial resolution is very high: Z J Z W; KJ Z U; J Y JH K they linearly reduce to 6 at 2 Fourier modes in4direction at : X O 6 " 6 I O X ; 9 Fourier modes in axial direction: I O XY Radial range divided in 2 (uneven) intervals Computational parameters P.Luchini & M.Quadrio SIMAI Ischia
19 \0 k x \0 k z 0 7 u v w 0 7 u v w 0 0 S ii (k] x ) S ii (k z ) d energy spectra P.Luchini & M.Quadrio SIMAI Ischia
20 r/delta uv ^ uv -uv /Re du/dr channel Turbulent stresses P.Luchini & M.Quadrio SIMAI Ischia
21 r/delta o_i/u_tau^ o_x o_r o_t 0.4 RMS vorticity fluctuations P.Luchini & M.Quadrio SIMAI Ischia
22 P.Luchini & M.Quadrio SIMAI Ischia Conclusions An efficient method for DNS of turbulent flows in cylindrical geometries has been presented, as an extension of the cartesian case The number of Fourier modes in4direction has been varied with azimuthal resolution is constant: significant benefits, so that the Turbulent flow in the annular pipe has been studied for the first time via DNS Preliminary observations on the effect of the transverse (concave and convex) curvature on the turbulence statistics have been reported
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