A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows

Transcription

1 A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows F. Fang a,, C.C. Pain a, I.M. Navon b, M.D. Piggott a, G.J. Gorman a, P. Allison a, A.J.H. Goddard a a Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, Prince Consort Road, London, SW7 2BP, UK. URL: b Department of Mathematics Florida State University Tallahassee FL, USA. Preprint submitted to Ocean modelling 21 April 2008

2 Abstract A novel Proper Orthogonal Decomposition (POD) model has been developed for a mesh adaptive ocean model, the Imperial College Ocean Model (ICOM) [1]. The aim in this paper is to exploit the stability of the above POD-Galerkin reduced order model for moderate Reynolds numbers which are often encountered in ocean modelling. The effect of truncated modes is evaluated when adaptive meshes are introduced in the POD-Galerkin model. The stability of POD-Galerkin model with the use of adaptive meshes is evaluated using the Munk gyre flow test case with a range of Reynolds numbers ( ). It is shown that some POD models using the L 2 norm become unstable when the Reynolds number is larger than 400. Stabilisation improvements can be achieved using the H 1 POD projector rather than the L 2 POD projector. The POD bases are constructed by incorporating gradients as well as function values in the H 1 Sobolev space [2]. The stability of numerical results can also be enhanced by increasing the number of snapshots and POD bases. Error estimation was used to judge quality of reduced order adaptive mesh models. keywords: reduced-order modelling; ocean model; finite element; unstructured adaptive mesh, POD 1 Introduction Developments in numerical ocean modelling have mainly focused on improving the resolution of numerical models over the last few decades. An important contribution to the accuracy of ocean models has involved the introduction of unstructured meshes. A review of the advantages of models using unstructured meshes is given in [3]. Unstructured, dynamically-adaptive mesh models can efficiently resolve global, basin, regional and small-scale flow structures (e.g., in moterate-reynolds number vortex flows). The adaptive mesh technique is recently introduced to 4D-Var data assimilation capable of producing a best estimate model solution by fitting a numerical simulation to observational data over both space and time [4, 5]. The use of unstructured meshes yields the ability to adapt meshes dynamically through time and optimise the accuracy of both the forward and inverse calculations. However, the major difficulty in the implementation of ocean modelling and 4D-Var data as- Corresponding author address: f.fang@imperial.ac.uk (F. Fang). 2

3 similation is the large dimensionality (typically in the range ), and hence incurs high memory and computational costs in computing the cost function and its gradient that require integration of model and its adjoint. To overcome this difficulty, reduced-order modelling is a powerful concept which enables a representation of the dynamics of large-scale systems on a small number of modes. Proper Orthogonal Decomposition (POD) methodology, in combination with the Galerkin projection procedure has been shown to provide an efficient means of generating reduced order models ([6, 7, 8]). The Galerkin projection on POD subspace transforms directly the PDE (Partial Differential Equation) system of the incompressible Navier-Stokes equations in a system of ODE (Ordinary Differential Equation), which require a very limited computational effort. This technique identifies the most energetic modes in a time-dependent system thus providing a means of obtaining a low-dimensional description of the system s dynamics. POD has been widely and successfully applied to numerous fields, including signal analysis and pattern recognition ([9]), fluid dynamics and coherent structures ([10, 11, 6, 12]) and image reconstruction ([13]). The practical utility of the POD approach was extended to large scale complex flow dynamics such as atmospheric and ocean modelling and the four-dimensional variational (4D-Var) data assimilation ([14, 15, 16, 7]). Furthermore, the POD approach has been incorporated in an unstructured mesh finite element ocean model ([1], [17]) which includes an ability to simultaneously resolve both small and large scale ocean flows (as they evolve), and improved representation of bathymetry/coastlines. However, especially for transitional and turbulent flows (i.e. moderate and high Reynolds number), the price of the low-dimensionality is indeed a lack of stability, which either restricts reduced order models to a narrow range of parameters or to a short-time integration. High Reynolds flows in the ocean contain motions with a broad range of scales, i.e. display a combination of organised or coherent structures (associated with the phase-averaged/spatially phase-correlated components that exhibit the most evident structure) and apparently disorganised or incoherent structures (associated with the random components). The energy transfer/interaction between the different coherent/inherent structure flows plays an important role in characteristics of high Reynolds number flows. Low-order truncation of the POD basis however, inhibits all transfers between the large and small (unresolved) scales of the fluid flow. As a consequence, there is a lack of dissipation in POD and the system may diverge. Higher values of the Reynolds number correspond to a larger amount of kinetic energy in the smaller scales, therefore more POD bases should be retained [18]. To improve the accuracy of POD-Galerkin models, the effect of these unresolved modes must be included to provide an insight into the turbulent energy. Various calibration methods have been developed to enhance the stability of POD-Galerkin models. One way to achieve stabilisation improvements is to add calibrated terms into the POD reduced order equations [19, 20, 21]. The calibration terms can include: eddy-viscosity functions of the POD model, constant or/and linear terms. These calibration terms are computed by minimising a cost functional defined as either (a) the difference between the amplitude coefficients predicted by the calibrated POD 3

4 and those from the POD; or (b) the weak constraint functional, where the constraints are calibrated POD equations and are enforced by introducing Lagrange multipliers or adjoint variables. Another class of approaches relies on the modification of the boundary conditions or of the pressure term [22, 18, 23]. Galletti et al. and Noack et al. [18, 24, 23] identified the pressure term at the outflow boundary as the primary source for the lack of dissipation. Different models for the pressure term have been proposed based on a pressure-poisson problem [18] or a least-squares procedure [23]. Herein, the effect of the truncated modes is first exploited when adaptive meshes are introduced in the POD models. To recover the effect of the truncated bases (usually the small scales), a dissipative term is directly included in the inner product evaluations. POD is defined in the H 1 Sobolev space rather than in L 2, that is, to incorporate gradients as well as function values in the definition of POD (See details in [2]). The stability of the POD-Galerkin model developed for an adaptive mesh finite element ocean model is evaluated in a Munk gyre tested with a range of moderate Reynolds numbers ( ). 2 Reduced order ocean model The proper orthogonal decomposition (POD) is an efficient way to reduced order modelling by identifying the few most energetic modes in a time-dependent system, thus providing a means of obtaining a low-dimensional description of the systems dynamics. A 3D dynamical flow model is generally written as: u = f(u, t, x), (1) t where f is a general function representing a 3D nonlinear flow dynamics (here, Navier-Stokes equation), u is a vector containing all variables to be solved (e.g., velocities, pressure and temperature etc.), t is time and x = (x, y, z) T represents the Cartesian coordinate position. To construct the discretised ocean model, the linear basis function N is chosen for the velocity components and non-geostrophic pressure, whilst quadratic basis function M is used for the geostrophic pressure. The variables to be solved can be expressed in the finite element form: N N N u x,y,z = u i N i, v x,y,z = v i N i, w x,y,z = w i N i, i=1 i=1 i=1 N p ng = p ng,i N i, N p g = p g,i M i, (2) i=1 i=1 where, N is the number of nodes, p ng and p g are the non-geostrophic and geostrophic pressures, respectively. 4

5 2.1 Proper Orthogonal Decomposition The model variables (e.g., u, v, w, p) are sampled at defined checkpoints during the simulation period [t 1,..., t n,..., t K ], also referred to as snapshots (K being the number of snapshots). The snapshots can be obtained either from a mathematical (numerical) model of the phenomenon or from experiments/observations. The sampled values of variables at the snapshot i are stored in a vector U i with N entries (N being the number of nodes), here, U can represent one of variables u, v, w, p. The average of the ensemble of snapshots is defined as: Ū i = 1 K K U k,i, 1 i N, (3) k=1 Taking the deviation from the mean of variables,forms V k,i = U k,i Ūi, 1 i N. (4) A collection of all V k,i constructs a rectangular N by K matrix A. The goal of Proper Orthogonal Decomposition (POD) is to find a set of orthogonal basis functions Φ = Φ 1, Φ 2,..., Φ K such that it maximises 1 K < V k, Φ k > L 2 2, (5) K k=1 subject to K < Φ k, Φ k > L 2 2 = 1. (6) k=1 where <, > L 2 is the canonical inner product in L 2 space. The Singular Value Decomposition (SVD) is used to find the optimal base Φ of the optimisation problem (5). From SVD, the matrix A R N K can be expressed A = X Λ 0 Y T, (7) 0 0 where, Λ = diag(σ 1, σ 2,..., σ d ) R d d, X R N N and Y R K K are the matrices which consist of the orthogonal vectors for AA T and A T A respectively. The order N for matrix AA T is far larger than the order K for matrix A T A. Therefore a K K eigenvalue problem is solved Cy k = λ k y k ; 1 k K, (8) where C = A T A. This procedure is equivalent to a Singular Value Decomposition (SVD). The eigenvalues λ k are real and positive and should be sorted in an descending order. The POD basis vectors Φ k associated with the eigenvalues λ k are orthogonal and expressed as follows: Φ k = Ay k / λ k. (9) 5

6 It can be shown [6,11], that the k th eigenvalue is a measure of the kinetic energy transferred within the k th basis mode (strictly speaking this is applied, when the field under consideration is the velocity field, but can be generalised to other fields as well). If the POD spectrum (energy) decays fast enough, practically all the support of the invariant measure is contained in a compact set. Roughly speaking, all the likely realisations in the ensemble can be found in a relatively small set of bounded extent. By neglecting modes corresponding to the small eigenvalues, the following formula is therefore defined to choose a low-dimensional basis of size M (M << K), Mi=1 λ i I(M) = Ki=1, (10) λ i subject to M = argmin{i(m) : I(M) γ}, (11) where, 0 γ 1 is the percentage of energy which is captured by the POD basis Φ 1,..., Φ m,..., Φ M. 2.2 POD-Galerkin reduced order model The variables in (1) can be expressed as an expansion of the POD basis functions {Φ 1,..., Φ M }: M u(t, x, y, z) = ū + α m (t)φ m (x), (12) m=1 where ū is the mean of the ensemble of snapshots for the variables u(t), α m (1 m M) are the time-dependent coefficients to be determined, and α m (0) are the coefficients at the initial time level. Substituting (12) into (1) and taking the POD basis function as the test function, then integrating over the computational domain Ω, ( ) M Φ m f (ū + α m Φ m (x)), t, x dω. (13) Ω m=1 The POD reduced model is then obtained: ( ) α m M = f (ū + α m (t)φ m (x)), t, x, Φ m, (14) t subject to the initial condition m=1 α m (0) = ((u(0, x) ū(x)), Φ m ). (15) In the finite element method, the POD basis Φ m (x) = N i=1 N i Φ m,i, equation (14) can therefore be written as: ( ) α m M N = f (ū + α m (t) N i Φ m,i ), t, x, Φ m, (16) t m=1 i=1 where N i is the basis function in the finite element and N is the number of nodes in the computational domain. 6

7 3 POD approach in an adaptive mesh ocean model 3.1 Imperial College Ocean Model and mesh adaptivity In this work, a POD-based reduced model is developed for ICOM that can simultaneously resolve both small and large scale ocean flows while smoothly varying resolution and conforming to complex coastlines and bathymetry [3, 25]. With more appropriate focused numerical resolution (e.g. adaptive and anisotropic resolution of fronts and boundary layers, and optimal representation of vertical structures in the ocean) ocean dynamics may be accurately predicted during future climatic change. The model consists of the 3-D non-hydrostatic Boussinesq equations. To accurately represent local flow around steep topography the hydrostatic assumption is not made in this work. Here, the pressure variable is split into the non-geostrophic and geostrophic parts which are solved separately. This allows the accurate representation of hydrostatic/geostrophic balance (for details see [25, 3]). The mesh adaptivity technique used in this model constructs a metric tensor which encodes local error estimates using information from the current solution fields [26]. The metric tensor is used to calculate the required edge lengths and orientation to control solution errors. It is constructed so that an ideal edge length is unity when measured in metric space. This gives a guide to those elements which have edges that are too large or too small. Since the metric is dependent on both location and direction it is able to reflect locally anisotropic information within the solution. Thus inhomogeneous and anisotropic meshes result from this approach. By defining an objective functional which is based on the element quality in this metric space, an optimisation technique is used to improve the overall quality of the mesh. The operations of adaptive meshes include: edge collapsing/splitting; face to edge and edge to face swapping; edge to edge swapping; and local node movement or mesh smoothing in a fashion similar to Freitag and Ollivier- Gooch [27] and Buscaglia and Dari [28]. Constraints are imposed on these operations so as to preserve the integrity of non-planar geometrical boundaries [26, 29]. 3.2 Mesh adaptive technique in POD When adaptive meshes are employed in ocean models, the mesh resolution requirements vary spatially and temporally, as the meshes are adapted according to the flow features through the whole simulation. The dimensional size of the variable vectors is different at each time level since the number of nodes varies during the simulation. Snapshots can therefore be of different length at different time levels. This unavoidably brings difficulties in the implementation of a POD-based reduced model for an adaptive model. To overcome these difficulties, a standard 7

8 reference fixed mesh is adopted for the reduced model. The solutions from the original full model are interpolated from their own mesh onto the same reference fixed mesh at each time level, and then stored in the snapshots. The information at the snapshots is used to find the optimal POD basis. This allows the same length of base modes to be obtained at each time level of the numerical simulation. The resolution of the reference mesh and the piecewise linear interpolation errors between the two meshes (the adaptive mesh and the fixed reference mesh) may affect the accuracy of the POD simulation. The effect of the interpolation error has been investigated using flows past a cylinder [1]. To reduce the interpolation error, a high order interpolation approach is suggested in future work. 3.3 Geostrophic pressure A deficit of dissipation in the energy budget, causing the POD-Galerkin model to drift could be related to the pressure terms which appear at the boundaries of the computational domain [23, 18]. To incorporate the pressure effects due to confinement, Noack et al. [23] and Galletti et al. [18] proposed a quadratic model (based on a Poisson problem) and a linear model (based on a least square procedure) for the pressure term respectively, Taking into account the role of the pressure term in both the POD-Galerkin model and the geostrophic balance in ocean modelling, the pressure in the momentum equations is divided into two parts: p = p ng +p g. To accurately represent geostrophic pressure its basis functions are split into two sets: Φ pgu and Φ pgv which are associated with the u- and v-velocity components. The geostrophic pressure can be obtained from a quadratic finite element representation whilst linear finite element representations are used for the velocity components. Furthermore the geostrophic pressure can be represented by a summation of the two sets of geostrophic basis functions, which are calculated by solving the elliptic equations using a conjugate gradient iterative method (for details see [1]). The numerical results are significantly improved by using the new numerical method described above. 4 Stabilisation of reduced order model using a Sobolev H 1 norm High Reynolds flows in ocean contain motions with a broad range of scales, i.e. display a combination of organised or coherent structures (associated with the phase-averaged/spatially phase-correlated components that exhibit the most evident structure) and apparently disorganised or incoherent structures (associated with the random components). The higher the Reynolds number, the broader this range of scales. The incoherent turbulence is typically of a 8

9 time scale considerably smaller than that of the coherent structures. The lifetime of a coherent structure seems to decrease with increasing Reynolds number, when a structure appears to undergo fairly rapid evolutionary change through complex interactions (like tearing and fractional and partial pairings) or decay via turbulent diffusion by incoherent turbulence [30]. The distorted or subdivided structures find ways via mutual interactions to re-emerge and generate a new coherent structure. It is also noted that coherent and incoherent flows are not independent even if they are uncorrelated. Coherent structures both produce and spatially organise incoherent turbulence. One can see that Reynolds shear stresses and the energy transfer/interaction between the different coherent/inherent structure flows play an important role in characteristics of high Reynolds number flows, i.e., the generation, evolution and decay of turbulent flows (e.g., eddies). It is not surprising that the POD reduced model derived using the Galerkin approach is not sufficiently accurate in reproducing the dynamics of high Reynolds number flows since the truncation applied in the POD subspace inhibits all transfers between the different scales of the fluid flow. The neglected POD modes correspond to small scale structures and introduce dissipative errors in the model. As a consequence, the system may lose its long-term stability. The stabilisation of reduced order model can be achieved by introducing an artificial dissipation by using a Sobolev H 1 inner product norm, that is, the derivatives of the snapshots as well as that of the basis functions are included in the definition of POD [2]. One seeks the POD basis function Φ = Φ 1, Φ 2,..., Φ K such that it maximises : 1 K K < V k, Φ k > H 1 2 = 1 k=1 K K N (V k,i Φ k,i ) 2 + ɛ 1 k=1 i=1 K K N ( V k,i Φ k,i ) 2, (17) k=1 i=1 subject to K < Φ k, Φ k > H 1 2 = 1, (18) k=1 where ɛ is a coefficient to be chosen from dimensional analysis considerations [2], one may guess the value of epsilon. The POD basis function Φ k can be calculated as in the L 2 case: where y k is obtained by solving the eigenvalue problem: Φ k = Ay k / λ k, (19) Cy k = λ k y k ; 1 k K, (20) where C is an K by K matrix donated by A T A and its gradient A T A: C = A T A + ɛ A T A. (21) 9

10 5 Application and discussion The stability of POD-Galerkin model with the use of adaptive meshes is evaluated using the Munk gyre flow test case. The effect of the truncated modes is evaluated within a range of moderate Reynolds numbers ( ). To improve the stabilisation of numerical results, the POD bases are defined in a H 1 Sobolev space. A comparison of results using the H 1 and L 2 POD results is carried out. 5.1 Description of the case: The Munk Gyre The POD-Galerkin reduced order model is tested in a computational domain, 1000 km by 1000 km with a depth of H = 500 m. The wind forcing on the free surface is given τ y = τ 0 cos(πy/l), τ x = 0.0, (22) where τ x and τ y are the wind stresses on the free surface along the x and y directions respectively, and L = 1000 km. A maximum zonal wind stress of τ 0 = 0.1 Nm 1 is applied in the latitude (y) direction. The Coriolis terms are taken into account with the beta-plane approximation (f = βy) where β = and the reference density ρ 0 = 1000 kgm 1. The problem is non-dimensionalised with the maximum Sverdrup balance velocity βhρ 0 v = τ y τ 0π L v m/s 1 (23) (and so the velocity scale U = m/s 1 is used here), and the length scale L = 1000 km. Time is non-dimensionalised with T = L. Incorporating the beta-plane approximation gives U a non-dimensional β = L2 β = The non-dimensional wind stress (applied as a body U force here averaged over the depth of the domain) takes the same cosine of latitude profile with τ0 = τ 0L = The time step is 3.78 (U 2 ρ 0 H) 10 4, equivalent to 3 hrs. No-slip boundary conditions are applied to the lateral boundaries. The spin-up period is (50 days). The Reynolds number is defined as Re = UL. ν The POD bases are constructed by the snapshots which are obtained from the numerical solutions by forcing the full model with the background flow. To evaluate the effect of truncated modes, a range of moderate Reynolds numbers ( ) was used. Various numbers of snapshots and POD bases (dependent on the values of Reynolds numbers) for the velocity u, v, w and pressure p are chosen to capture more than 99% of energy. An adaptive mesh is adopted in the full model. The mesh for the full model adapts every 10

11 19 time steps with maximum and minimum mesh size of 0.2 and (non-dimensional) respectively. To allow the same length of POD bases at the snapshots for the reduced order model, a reference fixed mesh is chosen. To build up the snapshots, the solutions from the full model are interpolated from the adaptive mesh onto the reference fixed mesh. The effect of the interpolation error has been investigated using flows past a cylinder [1]. The aim of this paper is to evaluate the effect of the truncated modes when adaptive mesh scheme are introduced in the POD bases. The POD-Galerkin model is derived using both the H 1 and L 2 norms. The root mean square error (RMSE) between the POD velocity solution and the true one at the time level n is used to estimate the error of the POD results: RMSE n = Ni=1 (U n i U n 0,i) 2 N (24) where, Ui n and Uo,i n are the vectors containing the POD velocity components and true ones at the node i respectively, N is the total number of nodes over the domain. 5.2 Results and discussion Test case I: Re = The case studies have been carried out with a range of moderate Reynolds numbers (Re = ). It is noted that the POD-Galerkin model with the L 2 norm can represent the velocity field well when Re 400. In the case of Re = 400, 30 POD bases and 41 snapshots are chosen for velocity u, v, w and pressure p, for which about 97% of energy is captured. The root mean square error (RMSE) of velocity results between the POD and full models is less than 3.5 m/s during the simulation period [50, 150] days. The POD-Galerkin model with the L 2 norm gradually lost its stability as the Reynolds number further increased (i.e., Re > 400). There are two ways tested here to stabilize the POD simulation: (1) increasing the number of POD bases and snapshots; (2) introducing an artificial dissipation, that is, defining the POD basis in a Sobolev H 1 inner product norm. For the cases (Re = 800, 1200), the number of snapshots was increased to 81 with 70 POD bases, where about 99.5% of energy is captured. It is found that the velocity results can be represented better using the H 1 norm (ɛ = for Re = 800 and ɛ = for Re = 1200) than those using the L 2 norm. The results from the POD reduced order model using the L 2 and H 1 norms are compared with those from the full model. The Root Mean Square Error (RMSE) of velocity results between the POD and full models during the simulation period is provided in figure 1. It can be seen that by using the H 1 norm, the RMSE of velocity results between the POD and full models was reduced by up to 25% at Re = 800 (figure 1 (a)) during the period t = days and 50% at Re = 1200 (figure 1 (b)) during the period t = days. The velocity contours for Re = 1200 are shown in figure 3. It is observed that the POD-Galerkin 11

12 model defined in the H 1 space is more dissipative and hence less subject to numerical instability than that defined in the regular L 2 space. Note that the coefficient ɛ value is directly related to the accuracy of solutions in the POD model defined in H 1 (figure 2). One may guess ɛ is a function of Reynolds number. Herein, the coefficient ɛ is 10 5 for Re = 800 and for Re = Test case II: Re = 2000 As the Reynolds number is increased further to 2000, the POD reduced order models defined in both the H 1 and L 2 spaces become oscillatory and unstable as the simulation time increases and the RMSE of velocity results increases to 9 m/s (figure 4 (a)). The instability of the POD model can be improved if the integration period is shortened by half. The POD model is derived based on the snapshots obtained during the shortened period [50, 100] days. The RMSE of velocity results between the POD and full models is reduced by a factor of 2 after the simulation period is shortened (figure 4 (b)). With an increase in the number of snapshots and POD bases defined in the H 1 Sobolev space, the RMSE of velocity is further decreased and remains a small value (less than 2 m/s). The error of velocity results between the full and POD reduced order model defined in both the L 2 and H 1 spaces is shown in figure 7. By using the H 1 norm and increasing the number of snapshots and POD bases, the error in the velocity field decreases by 30 50% in the larger part of flow (figure 7) while the correlation of velocity results increases up to 97% (figure 5). Figure 6 also shows the instability in the velocity results is removed by using the H 1 norm during the shortened period. A comparison of velocity results between the full model and POD model defined in H 1 (with 201 snapshots and 100 POD bases) is provided in figure 8 and the vorticity at time level t = 100 days is shown in figure 9. The velocity and vorticity results from the POD model exhibit an overall good agreement with those from the full model. [Fig. 1 about here.] [Fig. 2 about here.] [Fig. 3 about here.] [Fig. 4 about here.] [Fig. 5 about here.] [Fig. 6 about here.] [Fig. 7 about here.] 12

13 [Fig. 8 about here.] [Fig. 9 about here.] 6 Summary and conclusion The stability of POD-Galerkin model with the use of adaptive meshes is first investigated for high Reynolds flows using the Munk gyre flow test case. A wide range of Reynolds numbers ( ) is employed to evaluate the effect of the truncated modes when the adaptive scheme is employed in the POD basis. A comparison of results using the H 1 and L 2 POD- Galerkin models is carried out. The results showed that the stabilisation of the POD-Galerkin model can be improved by introducing an artificial dissipation using a Sobolev H 1 inner product norm when Re = The RMSE of velocity results between the full model and POD- Galerkin model is reduced by 10% 50% with the use of the H 1 norm. Both the velocity and vorticity results at the different time levels exhibit an overall good agreement with those from the full model. An increase in the number of snapshots and POD bases also leads to an improvement in the accuracy of the POD model (for example at Re = 2000, the RMSE of velocity results decreases to a small value (less than 2.5 m/s)). Future work will consider to add calibrated terms (e.g., eddy-viscosity terms) to the POD reduced order equations to account for unresolved fine-scale fluctuations in the POD [24, 31, 32]. These calibrated terms can be calculated by resolving appropriate constrained minimisation problems. Acknowledgements This work was carried out under funding from NE/C52101X/1. Prof. I. M. Navon would like to acknowledge support of NSF grants ATM and CCF managed by Dr. Eun K. Park, and NSF Award ID References [1] F. Fang, C. C. Pain, I. M. Navon, M. D. Piggott, G. J. Gorman, P. Allison, and A. J. H. Goddard. Reduced order modelling of an adaptive mesh ocean model. International 13

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15 for fluid flow modelling. J. Comput. Phys., 207:192220, [20] X. Gloerfelt. Compressible pod/galerkin reduced-order model of self-sustained oscillations in a cavity. 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference), pages Cambridge, Massachusetts, [21] B. Galletti, C. H. Bruneau, L. Zannetti, and A. Iollo. Accurate model reduction of transient. flows. Technical Report, 5676:INRIA, [22] D. Rempfer. Low-Dimensional modelling and numerical simulation of transition in simple shear flows. Annu. Rev. Fluid Mech., 35:22965:doi: /annurev.fluid , [23] B. R. Noack, P. Papas, and P. Monkewitz. The need for a pressure-term representation in empirical galerkin models of incompressible shear flows. Journal of Fluid Mechanics, 497:335363, [24] B. Galletti A. Bottaro,, C. H. Bruneau, and A. Iollo. Accurate model reduction of transient flows and forced wakes. Eur. J. Mech. B/Fluids, 26 (3):354366, [25] R. Ford, C. C. Pain, M. D. Piggott, A. J. H. Goddard, C. R. E. de Oliveira, and A. P. Umbleby. A nonhydrostatic finite-element model for three-dimensional stratified oceanic flows. Part I: Model formulation. Mon. Weath. Rev., 132(12): , [26] C. C. Pain, A. P. Umpleby, C. R. E. de Oliveira, and A. J. H. Goddard. Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations. Comput. Methods Appl. Mech. Engrg., 190: , [27] L. A. Freitag and C. Ollivier-Gooch. Tetrahedral mesh improvement using swapping and smoothing. Internat. J. Numer. Methods Engrg., 40: , [28] G. C. Buscaglia and E. A. Dari. Anisotropic mesh optimization and its application in adaptivity. J. Numer. Methods Engrg., 40: , [29] M. D. Piggott, C. C. Pain, G. J. Gorman, P. W. Power, and A. J. H. Goddard. h, r, and hr adaptivity with applications in numerical ocean modelling. Ocean Modell., 10(1 2):95 113, [30] A. K. M. F. Hussain. Coherent structures-reality and myth. Phys. Fluids, 26(10): , [31] J. Favier, L. Cordier, A. Kourta, and A. Iollo. Calibrated pod reduced-order models of massively separated flows in the perspective of their control. Proceedings of 2006 ASME Joint US-European Fluids Engineering Summer Meeting, Miami, Florida, USA, [32] J. Favier, L. Cordier, and A. Kourta. Accurate pod reduced-order models of separated flows. Under consideration for publication in Physics of Fluids,

16 List of Figures 1 RMSE of velocity results between the POD model and the full model (a) Re = 800 (40 snapshots with 30 POD bases); (b) Re = 1200 (80 snapshots with 60 POD bases). The solid line: with the use of the L 2 norm; The dashed line: with the use of the H 1 norm RMSE of velocity results between the full model and the POD model defined in H 1 with different ɛ values Contour of velocity at time levels (Re = 1200): (a), (b) t = 100 days; (c), (d) t = days (Left panel: with use of the H 1 norm; Right panel: with use of the L 2 norm) RMSE of velocity results between the POD model and the full model (Re = 2000) (a) the POD model is derived based on snapshots obtained during a long period [50, 150] (81 snapshots with 60 POD bases); (b) the POD model is derived based on snapshots obtained during a short period [50, 100] (The solid line: 81 snapshots with 60 POD bases; The dashed line: 81 snapshots with 70 POD bases; The dot-dashed line: 201 snapshot with 100 POD bases) Correlation of velocity results between the full model and the POD model defined in both the H 1 (solid line) and L2 (dashed line) (Re = 2000) Contours of velocity at time level t = 75 days (Re = 2000): (a) with the use of the H 1 norm; (b) with use of the L 2 norm (81 snapshots with 60 POD bases) Error in the velocity field from the POD reduced model (Re = 2000) at time levels: (a) (b) t = 62.5 days; (c) (d) t = 150 days; Left panel: with the use of L 2 ; Right panel: with the use of H Contours of velocity at time levels (Re = 2000): (a) t = 62.5 days; (b) t = 75 days; (c) t = 87.5 days; (d) t = 100 days (the thick lines: the full model; the thin lines: the POD model defined in H snapshots and 100 POD bases are chosen) Vorticity from the full and POD reduced order models (Re = 2000) at time level t = 100 days (a) Full model; (b) POD model defined in the H 1 space with 201 snapshots and 100 POD bases

17 (a) Re = 800 (b) Re = 1200 Fig. 1. RMSE of velocity results between the POD model and the full model (a) Re = 800 (40 snapshots with 30 POD bases); (b) Re = 1200 (80 snapshots with 60 POD bases). The solid line: with the use of the L 2 norm; The dashed line: with the use of the H 1 norm. 17

18 Fig. 2. RMSE of velocity results between the full model and the POD model defined in H 1 with different ɛ values. 18

19 (a) t = 100 days (b) t = 100 days (c) t = days (d) Re = days Fig. 3. Contour of velocity at time levels (Re = 1200): (a), (b) t = 100 days; (c), (d) t = days (Left panel: with use of the H 1 norm; Right panel: with use of the L2 norm). 19

20 (a) (b) Fig. 4. RMSE of velocity results between the POD model and the full model (Re = 2000) (a) the POD model is derived based on snapshots obtained during a long period [50, 150] (81 snapshots with 60 POD bases); (b) the POD model is derived based on snapshots obtained during a short period [50, 100] (The solid line: 81 snapshots with 60 POD bases; The dashed line: 81 snapshots with 70 POD bases; The dot-dashed line: 201 snapshot with 100 POD bases). 20

21 Fig. 5. Correlation of velocity results between the full model and the POD model defined in both the H 1 (solid line) and L2 (dashed line) (Re = 2000). 21

22 (a) (b) Fig. 6. Contours of velocity at time level t = 75 days (Re = 2000): (a) with the use of the H 1 norm; (b) with use of the L 2 norm (81 snapshots with 60 POD bases). 22

23 (a) t = 62.5 days (b) t = 62.5 days (c) t = 150 days (d) Re = 150 days Fig. 7. Error in the velocity field from the POD reduced model (Re = 2000) at time levels: (a) (b) t = 62.5 days; (c) (d) t = 150 days; Left panel: with the use of L2 ; Right panel: with the use of H 1. 23

24 (a) t = 62.5 days (b) t = 75 days (c) t = 87.5 days (d) Re = 100 days Fig. 8. Contours of velocity at time levels (Re = 2000): (a) t = 62.5 days; (b) t = 75 days; (c) t = 87.5 days; (d) t = 100 days (the thick lines: the full model; the thin lines: the POD model defined in H snapshots and 100 POD bases are chosen). 24

25 (a) Full model (b) POD model defined in the H 1 space Fig. 9. Vorticity from the full and POD reduced order models (Re = 2000) at time level t = 100 days (a) Full model; (b) POD model defined in the H 1 space with 201 snapshots and 100 POD bases. 25