Abstract: - Determining when a numerical simulation is fully converged is important in getting accurate and reliable results. In this study a new convergence ...
POD Convergence Criterion For Numerically Solved Periodic Fluid Flows MOHAMED H. AHMED AND THOMAS J. BARBER Mechanical Engineering Department University of Connecticut Storrs, CT 06268 USA
Abstract: - Determining when a numerical simulation is fully converged is important in getting accurate and reliable results. In this study a new convergence criterion based on extracting the energetic modes of a numerical solution, based on the Proper Orthogonal Decomposition method, is suggested. The POD convergence criterion is tested and applied to flow over a bluff body which has been solved numerically. The higher the mode number, the longer it takes to converge. The results show that the singular values alone can be used to judge convergence based on POD numerical simulations. Also, the suggested POD convergence criterion can be applied successfully and easily to numerical simulations. Key-Words: - Convergence, Pod, Numerical, Fluid, Periodic, Bluff
1. Introduction Periodic flows are those unsteady flows in which flow variables have repetitive behavior over time. Many engineering flow applications exhibit some sort of periodicity such as in turbomachinery, gas turbine combustors, internal combustion engines, etc. Periodic unsteadiness in the flow field can occur naturally, e.g., the shedding of recirculation eddies behind bluff bodies to form a von Karman vortex street and the growth of instabilities in a shear layer as in the case of jet flows. Periodic unsteadiness can also occur when forcing is imposed on one or more of the flow boundaries. Numerical simulation has become an essential tool used in solving problems in many scientific areas. In many cases, it is less expensive and time consuming when compared to an experimental approach. An analyst should be able to demonstrate that numerical solutions are fully converged. In such unsteady flow problems, judging if a solution has converged is necessary in determining the correct spectral content of the solution. Classically, a numerical simulation is “said to be converged” if a calculated convergence parameter is less than some small specified value. More details about convergence can be found in [1]. In the case of unsteady periodic flow problems, the residual is
most likely to be periodic. This periodic behavior may change in shape as the solution progresses. Even when the residual fluctuates periodically below the specified value, the pattern of fluctuations may change as the solution runs further. Therefore, for unsteady problems, there is a question about what should be the specified value to judge convergence. Ahmed and Barber [1] proposed a new convergence criterion called FFT (Fast Fourier Transform) convergence criterion. That convergence criterion proved to be applied easily and successfully on some problems. The amplitude value of the secondary frequency fluctuates if its value is below 30 % of the amplitude of the dominant frequency. The Proper Orthogonal Decomposition (POD) is a very efficient and powerful method for data processing. The method is also known as Karhunen-Loeve Decomposition, Principal Components Analysis and Singular Value Decomposition (SVD). For a set of data, POD proves to find the optimum subspaces. The method has been widely used in many applications such as image processing [2], reduced order modeling [3], structure dynamics [4], extracting the coherent structures [5] and more. The aim of this study is to examine a new application of the POD; examining convergence of
periodic fluid flows. This new convergence criterion is based on extracting the most energetic modes and examining them as the solution progresses. Also, studying the behavior of the POD modes as the solution progresses helps in evaluating the errors in the data set used to get the final POD modes. In this paper, the POD convergence criterion is applied to the problem of flow over a triangular bluff body. In this problem, periodic unsteadiness occurs naturally due to shedding of the fluid flow from a boundary surface.
[U] represents the change of each mode with time, while [S] represents the weight contribution of each mode. [V] contains the eigenvectors that represent the spatial distribution of each mode. In fact, the spatial modes hold the coherent structures of the flow variables and they are qualitative. The first mode can be written as [U1 S1 V1], where U1 is the first row of the U matrix and S is the first value of the S matrix and V1 is the first column of the [V]T . The number 1 after each symbol represents the number of the mode.
2. POD Method
3. POD Convergence Criterion
Assuming a function A(x,t) or discrete data set in a computational domain, A(x,t) can be expressed as
The suggested new convergence criterion is based on examining the most energetic modes of a numerical solution using the POD method. This convergence method is applied on one or more variables over the entire computational domain or over some selected points distributed throughout the computational domain. Data samples are collected over a number of time steps and the POD modes found and examined as the solution progresses in time. Ideally the POD convergence is reached if all the POD modes do not change as the numerical solution progresses. As mentioned above, the temporal modes extract the temporal behavior of the energetic modes. Each temporal mode may contain more than one frequency therefore it is not easy to use the temporal modes to judge convergence. Also, the V modes extract the spatial values of the data set. If the data are collected over many data points, it will not be easy to consider the V modes in convergence. The easiest way is to use the singular values to judge convergence based on POD. First, we have to show that the singular reaches convergence after the U and V modes or at the same time at maximum to use it as the only parameter in the POD convergence criterion. It is important to select the number of snapshots that represent the expected frequencies correctly. The sampling rate determines the maximum frequency that can be captured. The sampling rate and the number of snapshots determine the minimum frequency value that can be captured correctly [1].
M
A( x, t ) = ∑ ai (t ) Φ i ( x)
(1)
i =1
where x is the spatial coordinate and t is time, ai (t ) are the eigenvalue coefficients and Φ i ( x) are the eigenvectors. Using the Singular Value Decomposition method (SVD) the data set matrix can be rewritten as T
[ A] n×m = U n×n S n×m V m×m
(2)
where [U ]n×n and [V ]m×m are two orthogonal matrices, n is the number of snapshots and m is the number of data points in each snapshot. [ S ]n×m is a non-negative diagonal matrix arranged in descending order that represents the eigenvalues of the system. The diagonal values are called singular values. The singular values can be interpreted as the weight of contribution of each mode in the POD reconstruction. The S values are the square roots of the eigenvalues of AAT or AT A. The eigenvectors of AT A make up the column of [V] while the eigenvectors of AAT make up the columns of U. Assuming selecting the first N energetic modes and ignoring the rest, the data set [A] can be approximated as
[ A] n×m = [U ]n× N [ S ]N × N [V ]m× N T
(3)
4. Results 4.1 Bluff body example Bluff bodies may be used as flame holders to stabilize flames in flowing combustible mixtures in ramjets and after burners of turbojets. The flame is held by introducing a strong circulation zone in the wake of the bluff body. Frequently, flame holders can be represented as bodies of triangular cross section, with the apex pointing into the onset flow. The flow behind a triangular bluff body has been calculated using the UTNS code developed by Choi [6]. The unsteady Navier-Stokes equations were solved using different turbulence models. Madabushi et al. [7] used the same code to study six different cases of flow behind the triangular bluff body and found that the k-ε model with wall functions gave the best overall comparison with the measured centerline data. The bluff body is an equilateral triangular cylinder with each side being 40 mm. The computational domain is about 5 times the length of the triangular side upstream and and 20 times in the downstream direction. The height is three times the length of the triangular base. A nonuniform mesh of 17,600 grid points is used in the calculations. The no slip boundary condition is applied on the walls of the triangular bluff body. Stagnation temperatures and pressures values are specified at the inlet. At the exit, the static pressure is specified while extrapolating the other variables. The inlet velocity is about 31 m/s with inlet Reynolds number of 77,500 based on the triangle base. The solution starts by solving the steady state flow condition for 100 iterations to initialize the unsteady solution. A time step of 3.65×10-5 s was used for 5000 time steps. Fig.1 shows the instantaneous unsteady flow field of the nondimensional v-velocity (velocity value/speed of sound) where velocity fluctuations can be observed clearly behind the bluff body. Vortex shedding starts at one of the triangle base corners then circulates behind the base, before moving downstream.
4.1 POD convergence analysis Fig.2 shows the normalized Root Mean Square (RMS) of the u and v velocity variables used to judge convergence of the triangular bluff body
simulation. The RMS of the v velocity has an unusual behavior as it decreases to a minimum value and then increases before reaching a “periodic” behavior. The sudden change is attributed to the formation and shedding of the corner vortex off of the triangular bluff body. This is the main source of unsteadiness in this problem. Both plots suggest that convergence is reached after ts = 2500. Seven points in the computational domain are chosen to investigate the applicability of the POD convergence criterion, as shown in Fig.3. Table 1 lists the values of the physical locations of these points. The dominant frequency, related to vortex shedding off the bluff body, is approximately 214 Hz. This corresponds to a Strouhal number of 0.276, based on the triangular bluff body base length and inlet velocity. For each FFT calculation, the number of the data samples is 256 and the sampling rate is four times the time steps to minimize the cutoff frequency (26.8 Hz in this case) [1], while the number of FFT points (NFFT) is 211.
Fig.1 Instantaneous radial velocity distribution over the triangular bluff body
flow
POD is applied to each data set [A] that consists of 256 snapshots at 7 spatial locations. The maximum number of modes that can be generated using SVD is 7 modes. At each new sampling rate, the first snapshot is deleted and the snapshots then shifted before inserting the new snapshot. This procedure is the same as sweeping the data set as the solution progresses. Before applying the POD convergence criterion, we apply the POD analysis to the axial velocity (u) using the last segment of the data set that starts at the time step of 3977 as shown in Fig.4. Fig.4a shows a part of the first 4 temporal modes (U1 to U4). Each temporal mode, except the first one, U1, has a mean value very close to zero. U1 captures mainly the mean values of the data set and a part of the fluctuating signal. Fig.4b shows the change of the normalized singular values of the first 5 modes with the mode number. As shown, the value of the singular values decreases rapidly as the mode number increases. The normalized S4 value is just
of mode 4 has many frequencies compared with that of mode 1.
(a)
1
Table 1. Point locations in triangular bluff body computational domain.
0.9
Normalized rms
0.8
Point p1 p2 p3 p4 p5 p6 p7
0.7
0.6
0.5
0.4 0
1000
2000 3000 Time Step
4000
5000
1
x/d -3.44 -1.93 0.53 1.08 5.09 10.40 15.50
y/d -0.97 0.77 0.30 0.30 0.27 0.24 0.20
Notes Upstream Upstream Downstream Downstream Downstream Downstream Downstream
0.9
0.7
(a)
0.2
0.6
0.1 0
U
0.5
-0.1
U1 U2 U3 U4
0.4 -0.2
0.3 0
1000
2000 3000 Time Step
4000
5000
0
10
20
(b)
1
Normalized S
Normalized rms
0.8
30
40
0.8 0.6 0.4 0.2 0
50
1
2
Sampling rate
4
0.5
V
0 V1 V2 V3 V4 V5
-0.5 -1 -1.5
0
1
2
3
4
5
6
7
Ponit number
Fig.4 POD analysis of the last segment of the axial velocity data set (starting time step = 3977). Fig3. Spatial location of the seven points under consideration of the flow over bluff body. 0.0115 while for S5 it is 0.0017. Considering only the first 4 POD modes is very good in capturing all the energies of this data set. Fig.4c shows the values of V as a function of the point number. The absolute coefficient values of the V matrix are on the order of 1. For the same point, the V value changes from one mode to another. FFT is used to study the frequency contents of the U1 and U5 as shown in Fig.5. The dominant frequency that has the maximum energy of mode 1 is the shedding frequency while for mode 4 it is the first harmonic of the shedding frequency. The frequency content
5
(c)
1
Fig.2 Nondimensional normalized RMS of the: (a) u- velocity and (b) v-velocity.
3
Mode number
Fig.5 Frequency content of the last data set of the axial velocity (a) U1 (b) U4. The POD convergence criterion is used to investigate the convergence of flow over a triangular bluff body. Fig.6a shows the change of
8
(a)
1
Singular values
10
S1 S2 S3 S4 S5
0
10
-1
10
-2
10
-3
10 1000
1500
2000
Normalized amplitudes
2500
3000
3500
4000
4500
5000
4000
4500
5000
Nondimensional time step (b)
1
10
0
10
-1
A1 A2 A3 A4 A5
10
-2
10
-3
10 1000
1500
2000
2500
3000
3500
Nondimensional time step (c)
2
Normalized V
10
V1 V2 V3 V4 V5
0
10
-2
10
-4
10 1000
1500
2000
2500
3000
3500
4000
4500
5000
Nondimensional time step
Fig.6 POD convergence history of the axial velocity (a) S Values (b) Maximum amplitudes of the temporal modes (U), and (c) Absolute values of the shape modes (V) at point 5. (a)
0.8 0.6 0.4 0.2 0
0
500
(b)
1
Normalized Amplitude
1
Normalized Amplitude
the singular values of each mode as the solution progresses. The singular value of each mode increases or decreases before reaching a uniform value. Generally speaking, as the mode number increases, the time it takes the corresponding singular value to reach uniform values increases. It takes about 3500 time steps for S4 to reach convergence while it takes 4300 time steps for S5 to reach convergence. Fig.6b shows the convergence of the U modes as the solution progresses (We include only the maximum amplitude of each mode in this comparison). As shown, the amplitude value of each temporal mode reaches convergence before the singular values. The changes of the V matrix are studied as a function of the progress of the solution to study their convergence. For each point there is a number of history curves that correspond to the number of modes. We include here V1 to V5 of only point 5, as shown in Fig.6c. The V values change greatly before reaching uniform values. The value of V5 at point 5 has a clear fluctuation but its value is very small compared to the other modes. This same behavior of V5 is observed for points 3 and 4 as well. The values of V5 at these points are close to zero, as shown in Fig.4c of the last segment of the data set. The value of V5 reaches convergence at about 4400, which is very close to the convergence value of S5. The other convergence values of the V values are less or very close to that of the S modes. It can be concluded from this that the change of the singular values can be used as a measure of convergence of the flow variables in numerical simulations. POD analysis is applied to the radial velocity (v) in order to investigate the energetic modes. The data set used is the last segment of data. Fig.7a shows a part of the first 4 temporal modes (U1 to U4). Fig.7b shows the change of the normalized singular values of the first 5 modes with the mode number. As shown, the value of the singular values decreases rapidly as the mode number increases. The first two modes have the most energy of the data set. The first 3 modes can be used to represent the data set of the radial velocity. All the temporal modes [U] have mean values that differ from zero. Also, each mode has more than one frequency content. As an example, Fig.8 shows that mode 4 has 6 frequencies, which are the shedding frequency and its harmonics.
1000
Frequency (Hz)
1500
0.8 0.6 0.4 0.2 0
0
500
1000
1500
Frequency (Hz)
Fig.8 Frequency content of the last data set of radial velocity (a) U1 (b) U4 The POD convergence criterion is applied to investigate the convergence of the radial velocity. Fig.9a shows the change of the singular values of the modes with the time steps. S1, S2 and S3 reach convergence at about the same time (after 2700 time steps), while mode 4 reaches convergence after 3000 time steps and S5 reaches convergence after 3500 time steps. The convergence values of S2, S3 and S4 are close to the corresponding values of the axial velocity. S1 and S5 of the radial velocity reach convergence after the corresponding ones of the axial velocity. It is clear from the convergence values of U and V of the radial velocity, shown in Fig.9, that they reach convergence at about the
same time as the singular values. These values of convergence of the axial and the radial velocities are of the same order of magnitude as that calculated using FFT convergence criterion [1] for the same problem and the same data sets. (a)
1
Singular values
10
0
10
S1 S2 S3 S4 S5
-1
10
-2
10
-3
10 1000
1500
2000
1
Normalized amplitudes
2500
3000
3500
4000
4500
5000
4500
5000
Nondimensional time step (b)
10
0
10
A1 A2 A3 A4 A5
-1
10
-2
10 1000
1500
2000
2500
3000
3500
4000
Nondimensional time step (c)
2
Normalized V
10
0
10
V1 V2 V3 V4 V5
-2
10
-4
10
-6
10 1000
1500
2000
2500
3000
3500
4000
4500
5000
Nondimensional time step
Fig.9 POD convergence history of the radial velocity (a) S Values (b) Maximum amplitudes of the temporal modes (U), and (c) Absolute values of the shape modes (V) at point 5.
5. Conclusion In this paper, we expanded the applicability of the Proper Orthogonal Decomposition method. A proposed new convergence criterion based on the POD analysis has been investigated. The convergence criterion is based on extracting the energetic modes and observing them as the solution progresses. Convergence is reached if the values of the energetic modes do not change as the solution progresses. Examining the convergence of the numerical solution of the flow over a triangular bluff body tests the applicability of the POD convergence criterion. The U and V modes reach convergence at the same time or close to the convergence values of the singular values. Using only the singular values to judge convergence of the
numerically solved periodic fluid flows is believed to be sufficient. This analysis has the advantage of examining the behavior of the energetic POD modes as the solution progresses besides estimating the errors in each mode after stopping the numerical simulation and getting results. The results show that there are some fluctuations of the V values after reaching convergence when the V values are close to zero. The POD convergence criterion can be used after reaching convergence using other criteria. To reduce the number of POD calculations, we suggest using the POD convergence criterion discontinuously by skipping some time steps. The larger the number of points where data are collected the better judging convergence using POD. References: [1] Ahmed, M. H, and Barber, T. J., FFT Convergence Criterion For Unsteady CFD Codes, AIAA 2004-0738, 42nd AIAA Aerospace Sciences Meeting and Exhibit, 5 - 8 Jan 2004, Reno, Nevada [2] Rosenfeld, C., and Kay, A., Digital Picture Processing, Academic Press, New York, 1982. [3] Barber, T. J., Narayanan, S., and Dorobantu, M., Reduced Order Modeling of Large-Scale Unsteadiness in Shear Flows, AIAA 2002-1064, 40th AIAA Aerospace Sciences Meeting and Exhibit, 14-17 Jan 2002, Reno, Nevada. [4] Han, S., and Feeny, B., Application of Proper Orthogonal Decomposition to Structural Vibration Analysis, Mechanical Systems and Signal Processing, Vol.17, No.5, 2003, pp. 9891001. [5] Alfonsi, G., Restano, C., and Primavera, L., Coherent Structures of the Flow Around a Surface-Mounted Cubic Obstacle in Turbulent Channel Flow, J of Wind Engineering and Industrial Aerodynamics, Vol.91, 2003, pp. 495511. [6] Choi, D., A Navier-Stokes analysis of film cooling in a turbine blade, AIAA-93-0158, 31th AIAA Aerospace Science Meeting, Reno, NV, 1993. [7] Madabhushi, R. K., Choi, D., and Barber, T. J., Unsteady simulations of turbulent flow behind a triangular bluff body, 33rd AIAA/ASME/SAE/ ASEE Joint Propulsion Conference & Exhibit, Seattle, WA, July 6-9, 1997.
