- Research article
- Open Access
Reduced order model of flows by time-scaling interpolation of DNS data
- Tapan K. Sengupta^{1},
- Lucas Lestandi^{2}Email authorView ORCID ID profile,
- S. I. Haider^{1},
- Atchyut Gullapalli^{1} and
- Mejdi Azaïez^{3}
https://doi.org/10.1186/s40323-018-0119-2
© The Author(s) 2018
- Received: 3 April 2018
- Accepted: 25 September 2018
- Published: 4 October 2018
Abstract
A new reduced order model (ROM) is proposed here for reconstructing super-critical flow past circular cylinder and lid driven cavity using time-scaling of vorticity data directly. The present approach is a significant improvement over instability-mode (developed from POD modes) based approach implemented in Sengupta et al. [Phys Rev E 91(4):043303, 2015], where governing Stuart–Landau–Eckhaus equations are solved. In the present method, we propose a novel ROM that uses relation between Strouhal number (St) and Reynolds number (Re). We provide a step by step approach for this new ROM for any Re and is a general procedure with vorticity data requiring very limited storage as well as being extremely fast. We emphasize on the scientific aspects of developing ROM by taking data from close proximity of the target Re to produce DNS-quality reconstruction, while the applied aspect is also shown. All the donor points need not be immediate neighbors and the reconstructed solution has equivalent relaxed accuracy. However, one would restrain the range where the flow behavior is coherent between donors. The reported work is a proof of concept utilizing the external and internal flow examples, and this can be extended for other flows characterized by appropriate Re–St data.
Keywords
- Time-scaling
- Interpolation
- LDC
- Flow past a circular cylinder
- ROM
Introduction
High performance computing using DNS for complex flow problems provide insight into physical mechanism at prohibitive cost of data storage, as voluminous data are created to resolve small scales in both space and time. DNS of Navier–Stokes equation (NSE) to understand flow generates huge amount of data. The major challenges of big data are processing, storage, transfer and analysis. The central motivations here is to replace time/memory-intensive DNS for the model problems of flow past a circular cylinder and LDC. Similar attempts are recorded in [43, 47] and other references contained therein. Memory requirements of such instability mode-based ROM in [43] come down drastically, due to the requirement of storing only fewer coefficients of the SLE equations and initial conditions. Henceforth this reference will be called SHPG for brevity.
There are numerous efforts in developing ROM’s, e.g. via Koopman modes, as in [13, 38]; dynamic mode decomposition in [39]; POD-based analysis of Reynolds-averaged Navier–Stokes (RANS) in [28, 29, 50]. In [10], authors reported low-dimensional model for 3D flow past a square cylinder using solutions of NSE obtained by a pseudo-spectral approach. However, even using thousands of snapshots, the reconstruction error was of the order of \(30 \%\), indicating an exponential divergence between any model prediction and the actual solution outside the snapshot range. In [30], authors used fourth order finite difference scheme for spatial discretization of NSE in primitive variable formulation for time accurate simulation for POD analysis of the flow field. The time discretization used second order accurate, three-time level discretization method, which invokes a numerical extraneous mode. It was noted that with only four POD modes, the model without pressure term gives rise to important amplitude errors which cannot be compensated by an increase in the number of modes. In naive energy-based POD approaches, researchers calculate amplitude functions of POD representation by solving ODE’s derived from NSE by simplifying nonlinear and pressure terms. Iollo et al. [21] have shown that this approach is inherently unstable. Thereafter many stabilisation techniques have been proposed [5, 8, 11, 20] in the finite element framework. A survey on projection based ROM for parametric problems is proposed by Benner et al. [7]. POD-Galerkin continues to be an active field of research for fluid dynamics problems, it has lead to recent successful application to finite volume [24, 52, 53] in velocity–pressure formulation. Generally, this approach enters in the reduced basis framework popularized in the early 2000s [26, 34] which is presented in detail in Quarteroni et al. books [35, 36]. Authors in [31] have also used an adaptive approach to construct ROM with respect to changes in parameters, by first identifying the parameters for which the error is high. Thereafter a surrogate model based on error-indicator was constructed to achieve a desired error tolerance in this work. Recent work lead by Pitton and Rozza [32, 33] has focused on applying ROM to detect bifurcations in the context of fluid dynamics. To do so, they developed accurate ROM and evaluated steady state eigenvalues of these ROM linearized Navier–Stokes operator to detect bifurcations. Yet, it was shown in [23] that singular LDC flow requires extremely accurate numerical schemes due to very high sensitivity to numerical conditions. Consequently, in this paper, we rely on previously established bifurcation diagram (see Refs. [22, 47] for details) to bound the ROM domain.
Other approaches have been explored, in particular relying on interpolation instead of projection. Among them, discrete empirical interpolation method (DEIM) in [6, 12] has encountered widespread success with applications ranging from non-linear multiparameter interpolation to hyper reduction techniques. A new family of interpolation method parametric PDEs problems has been developed by Amsallem and Farhat in series of papers [1–3]. The Grassmann interpolation method relies on a series of projection from the Grassmann manifold of solutions onto flat vector spaces on which usual interpolation techniques can be used. Then the interpolated field is projected back onto the manifold. This approach has proved very successful for aeroelastic flows [2] and has also been combined with other ROM tools such as DEIM in [4]. Yet, this approach relies on a complex mathematical framework and requires careful tuning to be accurate, for instance choosing the projection origin point. These issues have been addressed in recent thesis by Mosquera [27] which also proposes alternative algorithms. These technical difficulties motivate the introduction of simpler, physics based interpolation methods such as the one proposed in this paper.
The flow governed by unsteady NSE presents the physical dispersion relation linking each length scale (wavenumber) with corresponding time scale (circular frequency). Thus, the ranges of time and length scales are important, even though a single St and Re are often used to describe the flow field. Multitude of length and time scales also are inherently noted in [23] via POD modes and multiple Hopf bifurcations for flow in LDC. The existence of such ranges assists in developing a ROM, when donor Re’s are in the same range, where the target Re resides. If one takes one or two donor points far from the range where target Re resides, the presented ROM will provide a reconstructed solution, still with acceptable accuracy. These aspects of multiple Hopf bifurcations and existence of ranges of Re is highlighted in the present research, apart from developing an efficient ROM for this model problem.
For a vortex dominated flow, the time scale is defined as \(\mathrm{St}\; (= fD/U_\infty )\), relating dominant physical frequency (f) with flow velocity, (\(U_\infty \)) and the length scale (D). However the flow does not display a single frequency, as one notices several peaks for both flows in Fig. 1. The time series of the vorticity data at indicated locations are shown in the left hand side frames. While the flow past a circular cylinder displays a single dominant peaks with side bands in the spectrum (shown on the right hand side frames), the flow inside LDC clearly demonstrates multiple peaks. This property has been explored thoroughly for the LDC in [22] to explain the roles of multiple POD modes.
The existence of unique St for a fixed value of Re, as embodied in Eq. (1) implies that employing simple-minded interpolation strategies like Lagrange interpolation, will display unphysical wave-packets in reconstructed solution, as the time scales are function of Re at the target. This is clearly demonstrated in Fig. 3. The proposed ROM tackles this issue with the time scaling technique that is presented in this article.
Governing equations and numerical methods
These same methods have been used earlier for validating and computing the respective flows in [47], SHPG for flow over cylinder and in [44, 48, 49] for flow inside the LDC. Here the simulations are performed in a fine grid, with \((1001 \times 401)\) points in the \(\xi \) and \(\eta \) directions for the flow past circular cylinder, and \((257 \times 257)\) points are taken for the LDC problem.
Need for time scaling
The proposed ROM aims at interpolating vorticity fields at a target Re (\(Re_t\)) from precomputed DNS at different donor Re’s. If Lagrange interpolation is used directly, then it will not work due to variation of St with Re. Even with close-by donor Reynolds numbers data, upon interpolation, will produce wave-packets for flow past a cylinder as shown in Fig. 3. In this figure, results are shown for \(\hbox {Re} = 83\), as obtained by DNS of NSE (shown by solid lines) and that is obtained by Lagrange interpolation of NSE solution donor data obtained for \(\hbox {Re} = 78\), 80, 86 and 90.
We have also noted in SHPG that the flow past a circular cylinder suffers multiple Hopf bifurcations (experimentally shown in [17, 54]) and in [48] for flow inside LDC and flow over cylinder. Hence the accuracy of reconstruction naturally demands that the target and donor Re’s should be in the same segments of Fig. 3, as the flow fields are dynamically similar. In Fig. 3, the equilibrium amplitude of disturbance vorticity are plotted as a function of Re for both the flows. The equilibrium amplitude refers to the value of the disturbance quantity, which settles down in a quasi-periodic manner, due to nonlinear saturation after the primary and secondary instabilities. Presence of multiple quadratic segments in Fig. 3, indicates multiple bifurcations originating at different Re’s. Thus, it is imperative that one identifies the target Re in the same segment of donor Re’s for DNS-quality reconstruction for flow past circular cylinder as in SHPG and for flow inside LDC in [22]. In each of these sectors of Re, the flow behaves similarly and the (St, Re)-relation is distinct. It is to be emphasized that the present sets of simulations are performed using highly accurate dispersion relation preserving numerical methods.
The physical frequency (f) varies slowly with Re and superposition of time-series of donor data causes beat phenomenon observed by superposition of waves of slightly different frequencies. Thus, the knowledge of variation of St with Re is imperative in scaling out f-dependence of donor data before Lagrange interpolation and this is one of the central aspects of the present work. After obtaining frequency-independent data at target Re, one can put back the correct f-dependence via its variation with Re at the target Reynolds number.
For the next two ranges, no explicit excitation is needed (i.e., \(A_0 = 0\)) to achieve a stable limit cycle. \(\mathtt {R_{II}} = [8660:9350]\) and \(\mathtt {R_{III}} = [9450:10{,}600]\) are ranges for which the amplitude (\(A_e\)) follows a square root law, these are however different because of the peculiar behavior of the flow in the vicinity of Re \(= 9400\), which indicates the onset of second Hopf bifurcation. Finally, \(\mathtt {R_{IV}} = [10{,}600:12{,}000]\) is difficult for interpolation, as one can see two branches in this range, one of which is unstable (U-branch) with respect to any miniscule vortical excitation, as opposed to the stable one (S-branch). The flow past cylinder is also divided in ranges as shown in Fig. 3b. The range of Re from 55 to 130 is subdivided according to the bifurcation sequences by: \(55 \le \mathrm{Re} \le 68\); \(68 \le \mathrm{Re} \le 78\); \(78 \le \mathrm{Re} \le 90\); \(90 \le \mathrm{Re} \le 100\) and \(100 \le \mathrm{Re} \le 130\). For example, to reconstruct solution for Re = 83, we have used data in the range of \(78 \le Re \le 90\) for the most accurate ROM.
Formulation and modeling of ROM
Scaling constant and base \(Re_{b}\) for different range of \(Re_{s}\)
\({{\varvec{Re}}}\) range | Scaling constant ( \(\varvec{n}\) ) | Basic \({\varvec{Re}}\) ( \({\varvec{Re}}_{{\varvec{b}}}\) ) |
---|---|---|
\(55{-}68\) | \(-\,0.49\pm 0.02\) | 60 |
\(68{-}78\) | \(-\,0.41\pm 0.02\) | 72 |
\(78{-}90\) | \(-\,0.37\pm 0.02\) | 80 |
\(90{-}100\) | \(-\,0.32\pm 0.02\) | 95 |
\(100{-}130\) | \(-\,0.28\pm 0.02\) | 110 |
The search for \(t_0\) is performed in such a way that the phases of both \(Re_b\) and \(Re_s\) match accurately. One should note that the effects of \(t_0\) are significant, despite the fact that it has a very small value. There are many ways to compute \(t_0\), but accuracy must be very high in estimating it. A specific way is to view the time series in the spectral plane and using the imaginary part of FFT to be used as the accuracy parameter, as described in the next subsection.
Computing the initial time-shift (\(t_0\))
Time-scaling ROM algorithm for discrete DNS data
- 1.
Perform the algorithm (Algorithm 1) on all signals, except the base donor signal, in order to scale their oscillations.
- 2.Perform Lagrange interpolation on the scaled donor signals at target \(Re_t\) for all discrete times \(t_i\).where \({\bar{\omega }}^\star \) is the target signal and \(l_s\) are the Lagrange interpolation polynomials.$$\begin{aligned} {\bar{\omega }}^\star (t_i)=\sum _{s \in \mathrm {donors}} {\hat{\omega }}_s(t_i)l_s(Re_t) \end{aligned}$$(10)
- 3.
Scale-back \({\bar{\omega }}^\star \) to the physical time with \({t}^\star =\frac{{t}-t_0(Re_{t})}{(Re_b/Re_{t})^n}\).
Time-shifting ROM applied to the LDC flow
As we have shown in [23], the main frequency of the LDC flow is nearly constant across large ranges of Re, as shown here in Fig. 5. Thus, the time-scaling procedures simplify to a time-shifting procedure with \(n=0\), resulting in \({t}_s={t}-t_0\) for the donor and target points, which have the same frequency in Fig. 5.
In Fig. 9, the interpolated vorticity contours for \(\hbox {Re} = 9600\) are compared with those computed directly from NSE to show that interpolation works globally in the flow field and not merely at chosen sampling points. In this flow field, the power law exponent is zero and the strength of the interpolation is in obtaining the initial time shift (\(t_0\)) obtained using Algorithm 1, obtained from the FFT of the donor point vorticity with respect to the baseline Re chosen.
In the following, we study the case of flow past a circular cylinder to show the efficacy of the proposed time-scaling algorithm used here. For this flow also one notices presence of multiple time scales, but with a predominant frequency characterized by St, which follows the power law given by Eq. (6), with nonzero power law exponent, n.
Time-scaled ROM applied to the flow past a cylinder
RMS error estimates of interpolation for \(\hbox {Re} = 83\)
Cases | Re of donor points | Error for interpolation using donor points |
---|---|---|
I | (78,80,86,90) | 0.0434535949140671049 |
II | (72,80,86,90) | 0.0438833300701889223 |
III | (68,80,86,90) | 0.0445922677374889012 |
IV | (55,80,86,90) | 0.0624577915198629291 |
V | (55,80,86,130) | 0.140945940261735560 |
VI | (55,68,72,86) | 1.3159752726807628345 |
VII | (55,68,72,130) | 8.52240911220835436 |
We draw the attention on error estimates provided in Table 2 for different combinations of donor Re’s. It is evident from the table that the best result is obtained when all four donor points are in the same segment of target Re, as in Case I. In Cases II to IV, we have taken the lowest Re, farther to the left with increase in RMS error, with lowering of the smallest donor Re. But in Case V, the extreme Re’s are chosen as 55 and 130, and yet the RMS error is acceptable, as two of the donor Re’s belong to the segment of target Re. In contrast, for the Case VI, only a single donor Re belongs to the same segment, resulting in RMS error increasing almost ten folds as compared to the Case V. The worst case (Case VII) occurs in Table 2, when all the donor Re’s are outside the target Re segment. This justifies the scientific basis of the adopted ROM keeping the various ranges of Re punctuated by various Hopf bifurcations shown in Fig. 3b.
In this method, \(\omega '\) is reconstructed using the identical procedure of interpolation after time-scaling and initial time-shift, using Eq. 8 applied directly on \(\omega \) obtained by DNS. Thus, this procedure even circumvents the need to use the time-consuming method of snapshots to obtain POD modes that is required for any POD based ROM e.g. POD-Galerkin, interpolated POD. Unlike the methods of solving SLE equations given in SHPG, proposed ROM in this paper requires storage of at most four DNS data sets in each segment for most accurate reconstruction. If one is willing to settle for lesser accuracy, then one can reduce the requirement of performing DNS for two Re only, in each segment of Fig. 3. Hence this ROM is not memory intensive and it is faster.
The case for \(\hbox {Re}= 83\) are shown in Fig. 11c, d, which compare the disturbance vorticity at the same two locations with DNS data. Once again, the reconstructed ROM solution is indistinguishable from the corresponding DNS data. Thus, it is evident that spectrum with multiple peaks can be handled by the presented approach of time-scaling with initial time-shift, utilizing the power law between Re with St.
Summary and conclusion
Here, we have proposed time-scaled ROM for reconstructing super-critical flow past circular cylinder and flow inside LDC using time-scaled Lagrange interpolation of vorticity data obtained by DNS for different donor data at Re’s, largely located in the neighbourhood of the target Re. In performing the interpolation, a time-scaling is performed following Eq. (8) along with an initial time-shift, as a direct consequence of (St, Re)-relations given in Eqs. (6) and (7).
The proposed method differ from the ROM based on instability modes in SHPG, with respect to speed, accuracy and generality of application. ROM reconstruction at a target Re is of DNS-quality, if all the donor points belong in the same Re subrange, identified by multiple Hopf bifurcations in Fig. 3a, for flow inside the LDC in the range \(8700 \le Re \le 12{,}000\) and in Fig. 3b for flow past a circular cylinder, in the range of \(55 \le Re \le 130\) and in Table 1.
Data requirement of present ROM is at most for four Re’s located in the same subrange. If one wants to perform ROM with only three Re’s, then the reconstructed data are of slightly lower accuracy, but of very acceptable quality (not shown here). The present procedure provides scientific and applied basis of ROM, depending upon the number and location of donor points of target Re. The formulation of this procedure does not require the introduction of sophisticated mathematical tools contrary to Grassmann manifold interpolation but rather focus on physics to enable accurate low order model.
In instability based ROM in SHPG, one stores only the coefficients of SLE equations. However, one needs to obtain optimal initial conditions for the stiff SLE equations and is restricted to use of first five POD or three instability modes. This is due to difficulty in finding optimal initial conditions for SLE equations and only three instability modes have been used in SHPG. In the present approach, one finds initial time-shift (\(t_0\)) for the donor vorticity data with respect to a base Reynolds number. This time shift can be obtained by FFT based approach as proposed here.
Present study opens the scope of data mining in computational fluid dynamics. DNS of NSE produces massive amount of data which can be used economically to predict flow behavior of dynamical systems dominated by single or multiple peaks in the spectrum. The proposed ROMs can be used at any arbitrary Re on demand, by the proposed ROM performed with limited number of DNS at neighbouring Re’s. The novel procedure proposed here has been tested for the internal flow inside a LDC and an external flow over a circular cylinder, as proofs of concept.
Acknowlegements
The authors acknowledge the support provided to the second author from the Raman-Charpak Fellowship by CEFIPRA which made his visit to HPCL, IIT Kanpur possible. This work reports partly the results obtained during the visit.
Competing interests
The authors declare that they have no competing interests.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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