Sea Grant College Program Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Project No. 2008-ESRDC-01-LEV Presented at the Grand Challenges in Modeling Simulation, Summer Simulation Multiconference (GSMS), Ottawa, ON, Canada, July 11 - 14, 2010.
SPECTRAL ELEMENT/SMOOTHED PROFILE METHOD FOR TURBULENT FLOW SIMULATIONS OF
WATERJET PROPULSION SYSTEMS
X. Luo, C. Chryssostomidis and G. E. Karniadakis
MITSG 10-19
Spectral Element/Smoothed Profile Method for Turbulent Flow SimulationsofWaterjet Propulsion Systems
Xian Luo*, Chryssostomos Chryssostomidis* and George Em Karniadakis*** Design Laboratory, MIT Sea Grant, MIT ** Division of Applied Mathematics, Brown University
Keywords: ship propulsion, CFD, high-order methods,immersive boundary method
AbstractWe have developed fast numerical algorithms [1] for flowswith complex moving domains, e.g. propellers in free-spaceand impellers in waterjets, by combining the smoothed pro-file method (SPM, [2, 3, 4]) with the spectral element method[5]. The new approach exhibits high-order accuracy with re-spect to both temporal and spatial discretizations. Most im-portantly, the method yields great computational efficiencyas it uses fixed simple Cartesian grids and hence it avoidsbody-conforming mesh and remeshing. To simulate highReynolds number flows, we incorporate the Spalart-Allmarasturbulence model and solve the unsteady Reynolds-averagedNavier-Stokes (URANS) equations. We present verificationof the method by studying the turbulent boundary layer overa flat plate. We show that both the eddy viscosity and velocityfields are resolved very accurately within the boundary layer.Having developed and validated our numerical approach, weapply it to study transitional and turbulent flows in an axial-flow waterjet propulsion system. The efficiency and robust-ness of our method enable parametric study of many caseswhich is required in design phase. We present performanceanalysis and show the agreement with experimental data forwaterjets.
1. INTRODUCTIONDesign optimization of waterjet using computational fluid
dynamics (CFD) tools will lead to more efficient designs thatare smaller and may alleviate cavitation problems. However,for problems with such complex moving 3D geometries asrotors and stators in waterjets, standard CFD tools are ineffi-cient due to the very large computational time and the com-plex meshes required. For many simulations of waterjets, po-tential flows are assumed with limited viscous corrections,e.g. based on a two dimensional integral boundary layer anal-ysis [6]. There have been some RANS solvers applied to wa-terjet simulations, but numerical simulations of the interac-tion between rotor and stator in a fully unsteady manner aretoo complicated and computationally expensive. So many as-sumptions have been made, e.g. the rotor and stator problemis decoupled and the flow is rotationally cyclic so that one canmodel a single blade passage only [7].
To this end, we aim to develop fast high-order algorithmsfor numerical simulations of flows with complex moving do-mains, based on fixed simple Cartesian grids. In this paperwe first review our numerical approach ([8]) where we com-bine the smoothed profile method (SPM, [2, 3, 4]) with thespectral element method [5]. SPM is similar to the immersedboundary method (IBM, [9]) as they both use a force distribu-tion to effectively approximate the boundary conditions andhence to impose the rigid-body constraints. However, with thespectral element discretization SPM leads to high-order accu-racy as SPM adopts a smooth indicator function in contrast tothe direct delta function used in IBM. Furthermore, the hy-brid methodology leads to high computational efficiency; it ismuch faster (typically 1000 times faster) than using the often-employed arbitrary Lagrangian Eulerian (ALE) for simula-tions in moving complex domains.
Waterjet pumps often operate in high Reynolds num-ber regime and the flow is turbulent, so we incorporatethe Spalart-Allmaras (SA) turbulence model and solve theURANS equations to account for the subgrid stresses. Weshow that the method resolves accurately the turbulent bound-ary layer over a flat plate at Reynolds numberRe= 107. Sub-sequently, we present full 3D flow simulations of a waterjetpropulsion system and perform a parametric study.
2. NUMERICAL METHODOLOGY2.1. Representation of moving bodies
SPM represents the moving bodies by smoothed profiles(or the so-called indicator functions), which equalunity in-side the moving domains,zero in the fluid domain, and varysmoothly between one and zero in the solid-fluid interfacialdomain. In [1] we proposed ageneralform, which is effectivefor anydomain shape such as a propeller, i.e.,
φi(x, t) =12
[
tanh(−di(x, t)
ξi)+1
]
, (1)
where index i refers to theith moving body (e.g., a singleblade of rotor or stator). Also,ξi is the interface thicknessparameter anddi(x, t) is thesigneddistance to theith mov-ing body with positive value outside and negative inside. Forsimple geometries (cylinders, ellipsoids, etc.)di(x, t) can beobtained analytically. However, for general complex shapes,such as impellers which can be represented by many sur-face point coordinates, spline interpolations are used to cal-
culatedi(x, t) andthusφi(x, t). A smoothly spreading indica-tor function is achieved by summing up the indicator func-tions of all theNp non-overlapping moving bodies:φ(x, t) =
∑Npi=1 φi(x, t).
Based on this indicator function, thevelocity fieldof themoving bodies,up(x, t), is constructed from the rigid-bodymotions of each moving domain:
φ(x, t)up(x, t) =Np
∑i=1
{V i(t)+ωi(t)× [x−Ri(t)]}φi(x, t),
(2)whereRi , V i = dRi
dt andωi arespatial positions, translationalvelocity and angular velocity of theith moving body, respec-tively. The total velocity fieldis then defined by a smoothcombination of both the velocity field of moving bodiesup
and the fluid velocity fieldu f :
u(x, t) = φ(x, t)up(x, t)+(1−φ(x, t))u f (x, t). (3)
We see that inside the moving domains (φ= 1), we haveu =up, i.e., the total velocity equals the velocity of the movingbody. At the interfaces (0< φ < 1), the total velocity changessmoothly from the propeller velocityup to the fluid velocityu f .
SPM imposes the no-penetration constraint on the surfacesof the simulated moving bodies. It can be shown (ref. [1]) thatimposing the incompressibility condition of the total velocity∇ ·u = 0 ensures the no-penetration surface condition(∇φ) ·(up−u f ) = 0, and vice versa.
SPM solves for the total velocity,u, in the entire domainD,including inside the moving domains, using the incompress-ible Navier-Stokes equations with an extra force density term,i.e.,
∂u∂t
+(u ·∇)u = −1ρ
∇p+ν∇2u+ fs in D (4a)
∇ ·u = 0 in D, (4b)
whereρ is the density of the fluid,p is the pressure field,ν isthe kinematic viscosity of the fluid,g is the gravity (and otherexternal forces on the fluid), and the fluid solvent is assumedto be Newtonian with constant viscosity for simplicity.
Herefs is the body force density term representing the in-teractions between the moving bodies and the fluid. SPM as-signs
R
∆t fsdt = φ(up−u) to denote the momentum change(per unit mass) due to the presence of the moving bodies.Thus, at each time step the flow is corrected by a momen-tum impulse to ensure that the total velocity matches that ofthe rigid domains within the moving domain, hence enforcingthe rigidity constraint.
2.2. Fully-discrete system: temporal and spa-tial discretizations
To numerically solve equations (4), we developed a high-order temporal discretization [1] instead of the original fully-explicit scheme ([2]). We introduced a semi-implicit treat-ment, using a stiffly-stable high-ordersplitting (velocity-correction) scheme [10]. In particular, the viscous term istreated implicitly and the order of the time integration schemeis up to 3rd. This choice enhances the stability and also in-creases the temporal accuracy of the original SPM implemen-tation.
The hydrodynamic forceFh and torqueQh on the movingbodies exerted by the surrounding fluid are derived from themomentum conservation. Specifically, the momentum changein the moving domains equals the time integral of the hydro-dynamic force and the external force, and hence:
Fhni =
1∆t
Z
Dρφn+1
i (u∗−unp)dx (5a)
Qhni =
1∆t
Z
Drn+1
i × [ρφn+1i (u∗−un
p)]dx (5b)
where the indicesn,n+ 1 refer to the solutions at differenttime steps,u∗ is the intermediate velocity field in the splittingscheme, andrn+1
i is the distance vector from the rotationalreference point on theith moving body to any spatial pointx.
For spatial discretization, we apply the spectral/hpelementmethod (see [5]). This hybrid method benefits from both fi-nite element and spectral methods: on one hand, for domainswith complex geometry, we can increase the number of sub-domains/elements (h-refinement) with the error in the numer-ical solution decaying algebraically. On the other hand, withfixed elemental size we can increase the interpolation orderwithin the elements (p-refinement) to achieve an exponen-tially decaying error, provided the solutions are sufficientlysmooth throughout the domain. The spectral element methodhas great advantages because of its dual path to convergence,e.g. convergence check without re-meshing. Furthermore, theuse of smooth profiles in SPM preserves the high-order nu-merical accuracy of the spectral element method.
The spectral/hpelement method allows us to accuratelyrepresent arbitrary fixed rigid boundaries of the flow domainwhile using SPM allows us to represent the moving/complexdomains, e.g. impellers and stators.
2.3. Turbulence modelingThe Spalart-Allmaras (SA) model [11] is a one-equation
model, which solves a transport equation for a viscosity-likevariableν, which may be referred to as the SA variable, i.e.,
∂ν∂t
+u ·∇ν = Cb1[1− ft2]Sν−[
Cw1 fw−Cb1
κ2 ft2
](
νd
)2
+1σ{∇ · [(ν+ ν)∇ν]+Cb2|∇ν|2}+ ft1∆U2 (6)
REFERENCES[1] X. Luo, M. R. Maxey, G. E. Karniadakis, Smoothed
profile method for particulate flows: Error analysisand simulations, Journal of Computational Physics 228(2009) 1750–1769.
[2] Y. Nakayama, K. Kim, R. Yamamoto, Hydrodynamiceffects in colloidal dispersions studied by a new effi-cient direct simulation, in: Flow Dynamics, Vol. 832 ofAmerican Institute of Physics Conference Series, 2006,pp. 245–250.
[3] Y. Nakayama, R. Yamamoto, Simulation method toresolve hydrodynamic interactions in colloidal disper-sions, Physical Review E 71 (2005) 036707.
[4] R. Yamamoto, K. Kim, Y. Nakayama, Strict simulationsof non-equilibrium dynamics of colloids, Colloids andSurfaces A: Physicochemical and Engineering Aspects311, Issues 1-3 (2007) 42–47.
[5] G. E. Karniadakis, S. J. Sherwin, Spectral/hpElementMethods for CFD, Oxford University Press, New York,1999.
[6] H. Sun, S. Kinnas, Performance prediction of cavitatingwater-jet propulsors using a viscous/inviscid interactivemethod, in: Transactions of the 2008 Annual Meeting ofthe Society of Naval Architects and Marine Engineers,Houston, Texas, 2008.
[7] S. Schroeder, S.-E. Kim, H. Jasak, Toward predictingperformance of an axial flow waterjet including the ef-fects of cavitation and thrust breakdown, in: First Inter-national Symposium on Marine Propulsors, 2009.
[8] X. Luo, C. Chryssostomidis, G. E. Karniadakis, Fast3d flow simulations of a waterjet propulsion system, in:Proc. International Simulation Multi-conference, Istan-bul, Turkey, July 2009.
[9] C. S. Peskin, The immersed boundary method, Acta Nu-merica 11 (2002) 479–517.
[10] G. E. Karniadakis, M. Israeli, S. A. Orszag, High-ordersplitting methods for the incompressible Navier-Stokesequations, J. Comp. Phys. 97 (1991) 414.
[11] P. R. Spalart, S. R. Allmaras, A one-equation turbulencemodel for aerodynamic flows, La Recherche Aerospa-tiale No. 1 (1994) 5–21.
[12] H. Eca, L., A. M., Hay, D. Pelletier, A manufacturedsolution for a two-dimensional steady wall-bounded in-compressible turbulent flow, International Journal ofComputational Fluid Dynamics 21 (2007) 175–188.
[13] C. L. Rumsey, Apparent transition behavior of widely-used turbulence models, International Journal of Heatand Fluid Flow 28 (2007) 1460–1471.
[14] J. Boussinesq, Th ´eorie de l ′Ecoulement tourbillant,Mem. Present ´es par Divers Savants Acad. Sci. Inst. Fr.23 (1877) 46–50.
[15] P. Spalart, C. L. Rumsey, Effective inflow conditions forturbulence models in aerodynamic calculations, AIAAJournal 45 (10) (2007) 2544–2553.
[16] R. W. Kimball, Experimental Investigations and Numer-ical Modeling of a Mixed Flow Marine Waterjet, PhD.thesis, Dept. of Ocean Engineering, Massachusetts In-stitue of Technology, June 2001.
[17] J. S. Carlton, Marine propellers and propulsion, Lon-don, U.K., Butterworth-Heinemann Ltd., second edi-tion, 2007.
[18] D. Xiu, G. E. Karniadakis, The Wiener-Askey polyno-mial chaos for stochastic differential equations, SIAMJ. Sci. Comput. 24 (2002) 619–644.
