6th International Conference on Computational Heat and Mass Transfer Proceedings of 6th ICCHMT May 18–21, 2009, Guangzhou, CHINA
241 NUMERICAL SIMULATIONS OF TURBULENT HELICAL FLOW IN A RANQUE-HILSCH VORTEX TUBE WITH DIFFERENT RANS CLOSURES AND SUB-GRID SCALES MODELS 1
A. Secchiaroli*, 2R. Ricci, 3S. Montelpare, 4V. D’Alessandro, 5G. Artipoli 1,2,3,4,5
Dipartimento di Energetica – Facoltà di Ingegneria Università Politecnica delle Marche
*Corresponding author: Ph: +39 071 220 4359 Fax: +39 071 220 4770
ABSTRACT Numerical analysis of the internal flow in a RanqueHilsch vortex tube has been conducted using a computational fluid dynamic model, closer to the real geometry of a commercial vortex tube, developed by Ricci et al. [11]. The flow field computation in a Ranque-Hilsch Vortex Tube is a challenging task for CFD and for this reason both RANS and LES approaches have been used in the simulation of the turbulent flow. Flow field equations have been solved by means of the commercial code FLUENT™ 6.3.26. and different turbulence closures were used to obtain a numerical solution of Navier-Stokes equations for this complex flow. In particular, axial-symmetric RANS simulations has been conducted using RNG k-ε and a linear RSM (Reynolds Stress differential Model) closures as in [11], while LESs have been performed using Smagorinsky and Germano-Lilly dynamic subgrid models. Results showed, that turbulence closure models choice is a crucial issue in the prediction of the flow field and the thermal separation in a RanqueHilsch Vortex Tube. In fact, different simulations exhibit some differences in the description of velocity components, and secondary vortex structures. Simulations performed by both techniques are validated using Richardson extrapolation procedure on different grids [9]. Moreover flow patterns in the device has been also investigated by means of the calculation of the Helical Flow Index (normalized helicity) [12].
NOMENCLATURE ui , u j , uk = velocity vector components [m/s] 2 2 k = turbulent kinetic energy [m /s ]
r = radial coordinate θ = azimuthal coordinate
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S = Favre filtered strain rate tensor magnitude [s-1] R = RHVT radius [m]
G C I = Grid convergence index R ij = Reynolds stress based on Favre filtering
SGS = sub-grid scale M = Mach number
G = filter function for LES E = total internal energy per unit mass [J/kg] C R = Smagorinsky’s constant V = elementary volume in the computation domain
Ψ = Helical Flow Index x = position vector y = spatial coordinate t = time Greek Symbols μ = dinamic viscosity [Pa s] ρ = density ε = turbulent dissipation rate [m2/s3]
Δ = filter width in LESs [m]
Ω = space domain of the simulation Subscripts h = hot outlet c = cold outlet T = turbulent in = inlet Superscript
f = Favre’s averaged variable in time or space
f = Reynolds’ averaged variable in time or space d = deviatoric part of a tensor i = isotropic part of a tensor
6th Intternational Conferencee on Compuutational Heeat and Masss Transfer INTRODUCTION Ranque-Hillsch vortex tuube (RHVT) is a device without w moving parrt that is useed as low cosst refrigeratoor. This tube is suppplied by com mpressed gaas through a set of nozzles, azzimuthally arrranged in order o to prodduce a very strongg swirl motioon for the innlet gas. Inccoming high pressuure flow is split s in two streams withh very different thhermal characcteristics, thaat leave the device from outlett sections arrranged on thhe opposite side s of the tube. The, so caalled, “Energgy Separatio on” or Hilsch” effectt occurs in the tube: thhe hot “Ranque-H stream flow wing near thee internal waall of the tubbe exits from one siide of the devvice while thhe cold one flowing fl along the axis a exit from m the other side s (figure 1). 1 The commerciall RHVT moodeled in this paper is a coounterflow Exair® ® 25 scfm, ass in [1], suppplied by comp pressed air and show wed in Figuree 2.
ggrid spacing as a in [8], so thhis variable has h been used d in thhe computatiion of GCI [99]. T Three-dimens sional grids hhave been ussed for LESs. In thhese simulattions the moodel is comp mposed of abbout 1 199000 volum mes with a m maximum vollume of 3.03E1 [m3]. The LES grids ffeatures are reeported in Taable 11 2 together with w the ressults of gridd independeence a analysis, whiile a sketch of a grid iss representedd in f figure 4. It is clearly show wed that increaasing number of c cells beyond d 199000 iis useless. Neverthelesss a m modified gridd (not reportted in Table 2) was usedd to r reduce numerrical diffusion due to skeewness and nonn in radial dirrection.Total number of cells o orthogonality c iss more than 267000 withh a maximum m cell volumee of 2 2.71E-11 [m3]. ] Mathematicaal Model: M N Numerical sim mulation of fflow inside a Ranque-Hillsch d deals with thee prediction oof the dynam mic behavior of o a h swirling,, unsteady, coompressible turbulent high t flow w.
F Figure 1 Total tempperature mapp obtained byy LES; periphheral flow (hhot) and interrnal flow (colld) are showeed Although th he vortex tubbe is a simple device, th he fluid dynamic effect e that produce p therrmal separattion is extremely complex c andd not compleetely understoood. A description of the priincipal explanation effoorts of Ranque-Hillsch effect can be founnd in [2],w while a complete review r of theoretical, experimentaal and numerical works w aboutt vortex tubees can be fouund in [3].
ANALYSIS AND M MODELLIN NG Geometriccal and Comp putational Model: M Several sim mplificationss have been n introducedd into geometricall model of thhe RHVT in nvolving the inflow and the hoot outflow seections: inlett section hass been modified following fo Skkye et al. [1]], while outllet hot section has been represeented as an axial a outflow w in the computational domainn. Computattional models are showed inn figure 3 and figuure 4. Diifferent computational grids havve been desiggned for num merical simulationss. RANS sim mulations gridds features to ogether with the reesults of griid independeence procedu ure are listed in Taable 1, whilee a grid skettch is presennted in Figure 3. Mean M total temperature t d difference beetween hot and colld exit calculaated in the siimulation, haas been chosen as the t most sennsible physicaal parameter to the
Figgure 2 RHVT commercial c m model used in this work: Exair® ® 25 scfm Furthermore strong tempeerature gradiient occurs in F i a v vortex tube either in thhe axial than n in the raddial d direction, duue to the fl flow compreessibility, heence d dynamic probblem is stroongly coupleed with therm mal o one. Due too its charactteristics, parrticular caree is r required in modeling m this flow, establisshing governning e equations, seetting solutioon techniques and selectting tuurbulence closure c models. The coomplete set of g governing equuations is reppresented byy Navier Stokkes’ o one, in whicch gravity eeffects are excluded, e strress teensor is relaated to straiin rate one by constituttive r relations for Newtonian N fluuids, and thermal flux vecctor iss expressed by b Fourier’s ppostulate (1)--(6).
6th International Conference on Computational Heat and Mass Transfer simple turbulent flows (like round jet, mixing layer, boundary layer). Moreover, in this kind of closure models, Reynolds stress tensor normal components are assumed to be isotropic.
Table 1 Characteristics of RANS grids and Richardson’s extrapolation results Grid Code
Number of cells
RANS1 RANS2 RANS3 RANS4
5000 2500 2055 1212
ΔThc0
ΔThc0
[K] 27.6 25 22.6 20
% error 9.4 % 18 % 27.2 %
GCI 4% 4% 1.2%
Thermophysical fluid properties are expressed by a third order polynomial function of temperature. Boundary conditions are expressed, following Skye [1], by imposing pressure, total temperature, velocity vector direction and mass flow rate values at the computational inlet, and pressure values at the outlets as in Table 3.
∂ρ ∂ + ( ρu j ) = 0 ∂t ∂x j
(1)
∂ ∂ ∂p ∂τ ( ρui ) + ( ρuiu j ) = − + ij ∂t ∂x j ∂xi ∂x j
(2)
∂φ ∂ ∂ ∂ ( ρ E ) + ⎡⎣u j ( ρ E + p)⎤⎦ = − j + ( uiτ ij ) ∂t ∂x j ∂x j ∂x j
(3)
⎛ ∂ui ∂u j ⎞ 2 ∂uk δ + − μ ⎜ ∂x j ∂xi ⎟⎟ 3 ∂xk ij ⎝ ⎠ ∂T φ j = −λ ∂x j p = ρ RT R = 287.05 [ J / kg K ]
τ ij = μ ⎜
Figure 3 Sketch of a RANS grid This assumption was found to be incorrect in several classes of flows like strongly swirling flows, flows with significant streamline curvature and flows in non circular ducts (in which the anisotropy of this components is responsible for secondary vortex structures) [6].
(4) (5) (6) (7)
No-slip and adiabatic conditions are set at the solid bounds. In the RANS approach the governing equations are written for the time averaged quantities [7]. Nevertheless in the case of compressible flows, it is useful to eliminate density turbulent fluctuations from mean motion equation using Favre’s (or massweighted) average [7]; this approach has been used in the present work too. Modeling several unclosed terms presents particular difficulties due to flow compressibility and is not yet completely achieved. Anyway, in this work, two turbulence models have been used: RNG k-ε, and RSM. The first is a version of the well-known k-ε model more suitable for the prediction of high swirling flows, and it is based on Boussinesque’s hypothesis. Following Pope [4], it can be shown that this assumption is verified only if the turbulence adjust rapidly to the mean straining of the flow i.e. only for
Figure 4 Sketch of a LES grid Thus, a more complex turbulence model, is required in the RANS simulations of RHVT flow. In this work a second order model like RSM (Reynolds Stress Model) is used. This model is obtained writing differential transport equation for each component of Reynolds stress tensor. Hence one additional transport equations for each component of the Reynolds’ stress tensor, one for the turbulent kinetic energy and one for the turbulent dissipation rate must be resolved. This means solving six more partial differential equations, respect to the RNG k-ε model, and obviously increasing computational cost. Anyway second order closures are more useful when the flow features are the result of the anisotropy of the Reynolds’ stress tensor normal components, like high swirling flows, and stress-induced secondary flows.
6th International Conference on Computational Heat and Mass Transfer Table 2 Characteristics of LES grids and Richardson’s extrapolation results Grid Code LES1 LES2 LES3 LES4
Number of cells 198816 114616 102240 60210
ΔT
0 hc
[K] 14.3 14.2 11.4 10.9
ΔT
0 hc
if ( x, t ) =
Pressure Inlet pin [Pa]
0.05% 10.5 % 3.5 %
700000
Hot Pressure Outlet ph [Pa]
101325 101325
Inlet mass flow rate [g/s] Velocity Components Ratio Vθ/Vr
300 3.233 5
This features are both presents in the Ranque-Hilsch Vortex Tube flows, hence the choice of RSM models is natural in this kind of simulations. Further details about the models used in our RANS simulations can be found in [4,5,6,7]. For the discretization of convective terms in mass, momentum and energy conservation equations a SOU [5] scheme has been used, while, discretization scheme used for k, ε and RSM equations is a QUICK [5]. In contrast to RANS approach, in a Large Eddy Simulation, problem formulation must be unsteady and three-dimensional, so neither geometrical nor analytical simplifications are possible. For a compressible flow, averaging operation in the space domain (filtering) is performed by using Favre filtering as expressed in (7) for a generic flow variable. A simplification for the expression of unresolved terms in LES equations can be obtained through finite volume discretization method using a filter function G ( x - y ) , defined by (8), i.e. stating the equivalence between filter width and grid width. Unresolved quantities like sub-grid scale stress tensor, deriving from filtering operation, can be expressed by using the, so called, subgrid models. In this work two subgrid models have been used: Smagorinsky and Germano-Lilly (dynamic).
ρ ( y , t ' ) G ( x - y , t ' )d 3 ydt '
y ∈V
(8)
(9)
y otherwise
In both models, a linear relation between the deviatoric part of the unresolved stress tensor and Favre’s filtered strain rate tensor is assumed (11) (as in Boussinesque’s hypothesis), while the isotropic part of the unresolved stress tensor is calculated by means of sub grid scale Mach number (13) and filtered pressure; turbulent viscosity is expressed by the (12).
Value
Cold Pressure Outlet ph [Pa] Inlet Total Temperature Tin [K] [K]
∫
⎧⎪ 1 G (x - y) = ⎨ V ⎪⎩0
GCI
Table 3 Boundary conditions Parameter
Ω×[ 0,T ]
ρ f ( y , t ' ) G ( x - y , t ' )d 3 ydt '
Ω×[ 0,T ]
% error 0.1 % 19 % 23 %
∫
τ ij
SGS
" " d i ≅ Rij = − ρ uk i u j = τ ij SGS + τ ijSGS
τ d ij
SGS
1 ⎛ ⎞ = 2 μT ⎜ Sjij − Sj kk δ ij ⎟ 3 ⎝ ⎠ 2 μT = ρ (C R Δ ) S
τ i ij
SGS
1 = − γ M sgs pδ ij 3
(10) (11) (12) (13)
Anyway, in the Smagorinsky model the constant is equal to 0.1, while in the second model the constant is dynamically calculated on the information provided by a double application of the filtering operation [5]. This allows the constant to vary in space and time over a wide range of values. In Large Eddy Simulations discretization scheme used for convective terms is a low diffusion MUSCL [5], while time integration has been performed with a second order accurate implicit scheme. Frictionless flux treatment has been performed by means of a Roe Flux scheme [5]. Due to the high computational cost, physical time simulated has been set equal to 20 [μs]. This time interval is probably large enough to allow that swirl and axial velocity components (the most important ones) reach a steady condition. Anyway steady condition has been evaluated by monitoring time history of the integral pressure on the central section of the RHVT (Figure5). Thus LES result have been obtained averaging on a number of time steps corresponding to the steady condition. All the simulations (RANS and LES) have been conducted using a workstation equipped with an Intel® Core™2 Quad Q6600, at 2.4 GHz processor.
6th International Conference on Computational Heat and Mass Transfer
Figure 5 Integral Pressure value versus time step number (LES Smagorinsky)
Figure 6 Streamlines(extract at the last time-step) inside RHVT –Smagorinsky SGS model
RESULTS AND DISCUSSION In this section radial profiles of several physical parameters like swirl velocity, axial velocity and static temperature are reported. Diagrams are related to a tube section located at 50 mm from the hot exit i.e. at about a half of the tube length. The resulting flow field inside the RHVT showed in Figure 8 consists of two coaxial and co-rotating helical structures: the hotter one streams near the solid boundaries while the colder one near the axis. Helical structures flow along the axis with a variable axial velocity as come out from Figure 6. Secondary vortex structures appear on the meridian plane due to Reynolds’ stress tensor normal component anisotropy. These structures have been correctly predicted by both LES and RANS (RSM) techniques. In particular results from RANS simulations revealed a big secondary vortex structure that take the whole tube axial length, figure 7, as in previous work [1], while LES instantaneous results showed a fragmented and non-symmetric vortex structure, figure 8. Mechanism of vortex formation and break-down at the interface between the two coaxial streams is similar to Kelvin-Helmoltz instability figure (9) [10]. All the mathematical models, used in this work, predicts that swirl velocity is the highest component (see figures 9-10) as reported in other works [3,8,13]. Only LESs and RSM closure are able to predict a radial profile of swirl velocity similar to a forced vortex near the axis and to a free-vortex near the wall, figure. 10. Moreover LESs and RSM approaches show a good agreement in the prediction of swirl velocity profile, nevertheless the maximum value is overestimated in the RSM approach.
Figure 7 Secondary vortex structures plotted in a color map of swirl velocity, in a section of RHVT (produced by RANS simulation, RSM model)
Figure 8 Secondary vortex structures plotted in a color map of x-vorticity, (instantaneous result produced by LES, Smagorinsky SGS model)
6th International Conference on Computational Heat and Mass Transfer HFI is a non-dimensional parameter, particularly useful in the analysis of Beltrami flow fields, defined as in (13) and varying between -1 and 1.
Ψ ( x, t ) =
u ⋅ (∇ × u ) u ∇×u
(13)
Figure 9 Static Temperature Contour (at the early stage of the simulation ) inside RHVT – Smagorinsky SGS model RNG k-ε model, as we expected, produced predictions with important differences respect to those obtained by RSM and LESs for the velocity field in a high swirling flows [8]. In particular swirl velocity values are underestimated and hence axial velocity is overestimated, by this model (see figures10-11). A complete comparison between velocity profiles simulated by the different turbulent models employed in this work are reported in figures 12 and 13 for several positions along the tube axis. In these figures is underlined that the most simple turbulent model is unable to predict correctly velocity profiles in high swirl conditions. There is not a considerable difference in the predicted swirl velocity profiles between the two sub-grid scales models used in LES during this work. Anyway, the Dynamic Model predicts axial velocity component values higher than Smagorinsky’s ones in the colder zone, while in the hotter region Smagorinsky’s model predicts higher values than Dynamic’s one (see figure 11). RANS simulations and LESs showed also different capability in the temperature field simulations, figure 14. The temperature values calculated by the two RANS models used are always higher than LESs’ ones. Moreover the temperature profile obtained with the Dynamic Sub-Grid Scales Models, in the central zone of the domain, is colder than Smagorinsky’s one. In order to improve the understanding of the instantaneous flow features a parameter usually calculated in the analysis of swirling flows for biomedical applications (hemodynamics) has been used in LES: the Helical Flow Index (HFI) [12].
Figure 10 Swirl Velocity radial profiles at 50 mm from the hot exit
Figure 11 Axial Velocity radial profiles at 50 mm from the hot exit Obviously Ψ=1 when the flow is helical and Ψ=0 if the flow is purely axial or circumferential. Different HFI contours on several planes inside the tube are reported in the figures 15 and 16. This analysis showed an instantaneous pure helical flow near the wall (in some radial planes), an almost purely axial flow near the tube axis and a hybrid motion in the rest of the domain in which 0
