May 7, 2009 - The probe is in the shape of a truncated cone with a flat nose opening for ... to the uncertainty in the cone angle, the nose diameter, and the ...
50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
17th 4 - 7 May 2009, Palm Springs, California
AIAA 2009-2285
Efficient Uncertainty Quantification for the Design of a Supersonic Pressure Probe Serhat Hosder∗ Missouri University of Science and Technology, Rolla, MO 65409 Luca Maddalena∗† California Institute of Technology, Pasadena, CA 91125 The Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) method utilizing Euler CFD simulations has been applied to the stochastic analysis of a pressure probe designed for Mach number measurements in three-dimensional supersonic flows with moderate swirl. The probe is in the shape of a truncated cone with a flat nose opening for total pressure measurement and includes four holes on the cone surface for static pressure measurements. The present stochastic CFD study is performed to provide information for further refinement of the supersonic pressure probe design in the presence of geometric uncertainties and to quantify the variation in the pressure measurements and the Mach number due to the uncertainty in the cone angle, the nose diameter, and the location of the static pressure port on the cone surface. Each uncertain parameter was modeled as a uniform random variable with a specified range and mean value based on the tolerances supplied by the manufacturer. The uncertainty information for various output variables obtained with a second order polynomial chaos expansion fell within the confidence interval of the Latin Hypercube Monte Carlo statistics. The second order NIPC required only 20 CFD solutions to obtain the uncertainty information, whereas the Monte Carlo simulations were performed with 1000 samples (CFD solutions), indicating the computational efficiency of the polynomial chaos approach. The relative variation in the Mach number due to the specified geometric uncertainty was found to be less than 1%. The sensitivity analysis obtained from the polynomial chaos expansions revealed that the Mach number is an order of magnitude more sensitive to the variation in the cone angle than the uncertainty in the other geometric variables. The stochastic results also showed that the uncertainty in the cone pressure measurement remains constant and minimum as long as the static pressure holes are located beyond 20% of the truncated cone surface length indicating the possibility for a smaller-size probe design by moving the static pressure ports further upstream on the cone surface.
I.
Introduction
Stochastic computational fluid dynamics (CFD) methods are crucial for solving fluid dynamics problems that involve uncertainties associated with the modeling of the real life conditions. Such uncertainties may originate from various sources for a CFD problem such as the operating conditions, the geometry, initial or boundary conditions fed into the solver as input. To quantify the uncertainty in the solution and to achieve a certain level of robustness or reliability in the final design, these uncertainties have to be propagated to the output quantities of interest with stochastic methods. In the non-deterministic modeling of high-fidelity CFD problems with multiple uncertain input variables, computational efficiency becomes an important factor in the selection of the uncertainty propagation method due to the expensive nature of the computational simulations. In addition to the efficiency, a certain level of accuracy (confidence) is also desired in the solution of the stochastic problems. The Point-Collocation Non-Intrusive Polynomial Chaos (NIPC)1, 2 , a method developed to address the accuracy and efficiency concerns in stochastic CFD problems, is used in the ∗ Assistant
Professor of Aerospace Engineering. Member AIAA Scholar in Aeronautics. Graduate Aeronautical Laboratories. Member AIAA
† PostDoctoral
1 of 13 Copyright © 2009 by S. Hosder and L. Maddalena. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
current study to quantify the uncertainty in pressure and Mach number measurements in supersonic flows performed with a pressure probe in the shape of a flat-faced truncated cone with geometric uncertainty due to the manufacturing tolerances. The present stochastic CFD study is also performed to provide valuable information for further refinement of the supersonic pressure probe design in the presence of geometric uncertainties. Several methods, such as the interval analysis, propagation of uncertainty using sensitivity derivatives, Monte Carlo simulations, moment methods and polynomial chaos are the main approaches implemented in stochastic CFD simulations. Detailed description and analysis of each method for selected fluid dynamics problems can be found in Walters and Huyse.3 The method used in this paper is based on polynomial chaos (PC) , which is a method for the spectral representation of the uncertainty. Several researchers have studied and implemented the PC approach for a wide range of problems. Ghanem and Spanos4 (1990) and Ghanem5, 6 (1999) applied the PC method to several problems of interest to the structures community. Mathelin et al.7 studied uncertainty propagation for a turbulent, compressible nozzle flow by this technique. Xiu and Karniadakis8 analyzed the flow past a circular cylinder and incompressible channel flow by the PC method and extended the method beyond the original formulation of Wiener9 to include a variety of basis functions. In 2003, Walters10 applied the PC method to a two-dimensional steady-state heat conduction problem for representing geometric uncertainty. A comprehensive review of various polynomial chaos techniques in CFD is recently presented by Najm.11 The Polynomial chaos (PC) method for the propagation of uncertainty in computational simulations involves the substitution of uncertain variables and parameters in the governing equations with the polynomial expansions. In general, an intrusive approach will calculate the unknown polynomial coefficients by projecting the resulting equations onto basis functions (orthogonal polynomials) for different modes. As its name suggests, the intrusive approach requires the modification of the deterministic code and this may be difficult, expensive, and time consuming for many computational problems such as the one considered in this paper. To overcome the inconveniences associated with the intrusive approach, collocation-based NIPC methods have been developed for uncertainty modeling and propagation, such as the Point-Collocation1, 2 approach to be used in the current study and the Probabilistic Collocation approach developed by Loeven et al.12 The other NIPC approaches in the literature are based on sampling (Debusschere et al.,13 Reagan et. al.,14 and Isukapalli15 ) or quadrature methods (Debusschere et al.13 and Mathelin et al.16 ) to determine the projected polynomial coefficients. In a recent work by Eldred and Burkardt,17 a detailed description of various non-intrusive polynomial chaos approaches are given along with comparisons with the stochastic collocation methods for uncertainty quantification. In addition to the Point-Collocation NIPC method, Latin Hypercube Monte Carlo simulations with 1000 samples have been performed for uncertainty propagation in the present study. Both methods utilized Euler CFD simulations to numerically solve the flow field around the pressure probe geometries each corresponding to a sample point in the random domain. The results obtained with both methods are compared to show the computational efficiency and the accuracy of the Point-Collocation NIPC technique for the current stochastic CFD study. The paper is organized as follows: the next section gives a brief description of the Point-Collocation Non-Intrrusive Polynomial Chaos Method. The definition of the stochastic problem including the pressure probe details, geometric design parameters considered in the stochastic problem, and the description of the CFD simulations are given in Section III. Section IV presents the results obtained with the Latin Hypercube Monte Carlo and the Point-Collocation NIPC methods. The conclusions are given in Section V.
II.
The Point-Collocation Non-Intrusive Polynomial Chaos Method
The polynomial chaos is a stochastic method, which is based on the spectral representation of the uncertainty. An important aspect of spectral representation of uncertainty is that one may decompose a random function (or variable) into separable deterministic and stochastic components. For example, for any random variable (i.e., α∗ ) such as Mach number, density or pressure in a stochastic fluid dynamics problem, we can write, P X ~ = ~ α∗ (~x, ξ) αi (~x)Ψi (ξ) (1) i=0
2 of 13
~ is the random basis function corresponding to the where αi (~x) is the deterministic component and Ψi (ξ) ith mode. Here we assume α∗ to be a function of deterministic independent variable vector ~x and the ndimensional standard random variable vector ξ~ = (ξ1 , ..., ξn ) which has a specific probability distribution. In Equation 1, the discrete sum is taken over the number of output modes, P +1=
(n + p)! n!p!
(2)
which is a function of the order of polynomial chaos (p) and the number of random dimensions (n). The basis function ideally takes the form of multi-dimensional Hermite Polynomial to span the n-dimensional random space when the input uncertainty is Gaussian (unbounded), which was first used by Wiener9, 18 in his original work of polynomial chaos. Legendre (Jacobi) and Laguerre polynomials are optimal basis functions for bounded (uniform) and semi-bounded (exponential) input uncertainty distributions respectively in terms of the convergence of the statistics. The detailed information about polynomial chaos expansions can be found in Walters and Huyse3 and Hosder et al.1 To model the uncertainty propagation in computational simulations via polynomial chaos with an intrusive approach, all dependent variables and random parameters in the governing equations are replaced with their polynomial chaos expansions. Taking the inner product of the equations, (or projecting each equation onto k th basis) yield P + 1 times the number of deterministic equations which can be solved by the same numerical methods applied to the original deterministic system. Although straightforward in theory, an intrusive formulation for complex problems can be relatively difficult, expensive, and time consuming to implement. To overcome such inconveniences associated with the intrusive approach, non-intrusive polynomial chaos formulations have been considered for uncertainty propagation such as the method considered for the current study. The collocation based NIPC approach starts with replacing the uncertain variables of interest with their polynomial expansions given by Equation 1. Then, P + 1 vectors (ξ~i = {ξ1 , ξ2 , ..., ξn }i , i = 0, 1, 2, ..., P ) are chosen in random space for a given PC expansion with P +1 modes and the deterministic code (Navier-Stokes solver for the current study) is evaluated at these points. With the left hand side of Equation 1 known from the solutions of deterministic evaluations at the chosen random points, a linear system of equations can be written: α0 α∗ (ξ~0 ) Ψ0 (ξ~0 ) Ψ1 (ξ~0 ) · · · ΨP (ξ~0 ) Ψ0 (ξ~1 ) Ψ1 (ξ~1 ) · · · ΨP (ξ~1 ) α1 = α∗ (ξ~1 ) (3) .. .. .. .. .. .. . . . . . . Ψ0 (ξ~P )
Ψ1 (ξ~P ) · · ·
ΨP (ξ~P )
αP
α∗ (ξ~P )
The spectral modes (αk ) of the random variable, α∗ , are obtained by solving the linear system of equations given above. Using these, mean (µα∗ ) and the variance (σα2 ∗ ) of the solution can be obtained by µα∗ = α0
σα2 ∗ =
P X
D E ~ αi2 Ψ2i (ξ)
(4)
(5)
i=1
In 2006, Hosder et. al.1 applied this Point-Collocation NIPC method to stochastic fluid dynamics problems with geometric uncertainty. They demonstrated the efficiency and the accuracy of the NIPC method in terms of modelling and propagation of an input uncertainty and the quantification of the variation in an output variable. In that study which included a single random input variable, the collocation locations were equally spaced in the random space. Following that work, Hosder et. al.2 applied the Point-Collocation NIPC to model stochastic problems with multiple uncertain input variables having uniform probability distributions and investigated different sampling techniques (Random, Latin Hypercube, and Hammersley) to select the optimum collocation points. The results of the stochastic model problems showed that all three sampling methods exhibit a similar performance in terms of the accuracy and the computational efficiency of the chaos expansions. However, 3 of 13
xc
€ (a) Schematic of the probe
DN
δc
(b) Uncertain geometric variables used in the analysis
€
Figure 1. The pressure probe geometry used in the stochastic CFD analysis (dimensions in inches).
€
the convergence of the Point-Collocation NIPC statistics obtained with Hammersley and Latin Hypercube sampling exhibit a much smoother (monotonic) convergence compared to the cases obtained with random sampling. The solution of linear problem given by Equation 3 requires P + 1 deterministic function evaluations. If more than P + 1 samples are chosen, then the over-determined system of equations can be solved using the Least Squares method. Hosder et. al.2 investigated this option by increasing the number of collocation points in a systematic way through the introduction of a parameter np defined as np =
number of samples . P +1
(6)
In the solution of stochastic model problems with multiple uncertain variables, they have used np = 1, 2, 3, and 4 to study the effect of the number of collocation points (samples) on the accuracy of the polynomial chaos expansions. Their results showed that using a number of collocation points that is twice more than the minimum number required (np = 2) gives a better approximation to the statistics at each polynomial degree. This improvement can be related to the increase of the accuracy of the polynomial coefficients due to the use of more information (collocation points) in their calculation. The results of the stochastic model problems also indicated that for problems with multiple random variables, improving the accuracy of polynomial chaos coefficients in NIPC approaches may reduce the computational expense by achieving the same accuracy level with a lower order polynomial expansion. Based on the observations from previous studies by Hosder et. al.1, 2 performed with Point Collocation NIPC method for various stochastic fluid dynamics problems, a number of Latin Hypercube samples that is twice more than the minimum number required (2 × (P + 1)) are used as the collocation points for the current study.
III. A.
The Description of the Problem
The Pressure Probe
Figure 1(a) shows the schematic of the pressure probe geometry being used in the current study. The probe is designed in Caltech for Mach number measurements in three-dimensional supersonic flows with moderate swirl. The design is based on a truncated, conical tip that transitions to a cylinder. The cone has a half-angle of 10o and the cylindrical body has a diameter of 0.062500 . The four static pressure ports are located normal to the cone surface at a distance of 0.09500 from the truncated tip and have a diameter of 0.01200 . The static pressure ports lead to a common plenum, “Plenum A”, and are used to measure the static pressure on the cone surface, pc . The central port with a diameter of 0.1800 is used to measure the total pressure, p02 which is the value of the total pressure behind the normal shock that will stand in front the nose at supersonic speeds. By measuring cone-static and total pressure simultaneously, one can obtain the Mach number at the probe 4 of 13
location using the Krasnov’s similarity law19 which gives the functional relation between the Mach number and the total and cone pressures measured with the probe (M1 = f (pc /p02 )). The detailed description of the probe design, the calibration procedure, and obtaining the functional relation between the Mach number and the measured pressures are given in Maddalena et. al.20 In the current study, the uncertainty analysis has Cp been focused on the quantification of the variation in the measured Mach number due to the geometric uncertainty of the probe that may originate from the manufacturing tolerances. The main geometric design parameters considered in the uncertainty analysis (Figure 1(b) are the cone angle (δc ), the nose diameter (DN ), and the the distance of the static ports from the tip of the probe (xc ). Based on the information obtained from the probe manufacturer, the cone angle was modeled as a uniform random variable between 9.5o and 10.5o with a mean of 10o , the nose diameter was modeled as a uniform random variable between 0.01700 and 0.02700 with a mean of 0.02200 , and the static port distance measured on the cone surface from the nose xc was modeled as a uniform random variable between 0.09000 and 0.1000 with a mean value of 0.09500 . In addition to quan(a) Pressure contours from Euler solution tifying the uncertainty in the measured Mach numCp ber due to manufacturing tolerances, the stochastic results presented in this paper will provide useful information for studies that may target further refinement of the probe design. The non-intrusive polynomial chaos approach used in the present study will utilize CFD simulations to numerically simulate the flow field at a nominal Mach number of 2.0 at zero degrees angle of attack around a number of probe geometries each resulting from a unique set of geometric parameters in random space. Due to the uncertainty in the geometry, the measured total and static pressures would vary and create the uncertainty in the Mach number obtained with the Krasnov’s similarity law. For the present stochastic study, one can L think of the measured uncertain Mach number as M1 = M1 (ξ1 , ξ2 , ξ3 ) and each uncertain geometric parameter as a function of the component of € the standard random variable vector, δc = δc (ξ1 ), DN = DN (ξ2 ), and xc = xc (ξ3 ). 0.08L
B. (b) Pressure contours from full Navier-Stokes solution and the streamline pattern € in the separation region close to the nose
CFD Simulations
In this study, uncertainty quantification with polynomial chaos and Monte Carlo approaches used the Figure 2. The pressure coefficient contours and the solutions of Euler calculations performed with the flow structure around the probe defined by the mean CFD Code GASP ,21 which is a three-dimensional, geometric parameters structured, finite volume RANS code capable of solving steady-state or time-accurate flow problems. All CFD simulations were performed at a freestream Mach number of 2.0 at zero degrees angle of attack enabling axis-symmtric flow assumption. For the CFD simulations, Roe-Harten flux-difference splitting scheme was used to calculate the inviscid fluxes. The flux terms at the cell faces were calculated 5 of 13
using the upwind-biased third-order accurate MUSCL scheme with Spekreijse-Venkat limiter. Figure 3 shows the grid topology used in the Euler CFD simulations, which consists of three blocks. Three grid levels were used for each CFD simulation by grid sequencing to accelerate the convergence to the steady state solution. For each CFD simulation, the solution obtained with the finest grid level was used in the uncertainty analysis. At the finest mesh level, first block includes 53 × 25 grid points, second block has 53 × 157 grid points and the third block consists of 197 × 157 grid points. Figure 2 shows the typical flow field around the pressure probe at a free-stream Mach number of 2.0. In this figure, the probe geometry is defined by the mean values of the cone angle, nose diameter, and the static port distance from the nose. In addition to the Euler CFD, a Full-Navier Stokes simulation with the Spalart-Allmaras turbulence model was also performed for this case to study the difference between the flow structures obtained with the inviscid and viscous flow model assumptions. As expected for a blunt-body in supersonic flow, a bow shock stands in front of the probe. The portion of this 3
wave in the nose region is close to a normal shock inducing subsonic flow between the tip and the shock wave. As the flow 2
accelerates over the edge of the nose, it is compressed through another shock originating from the tip of the nose. It should be noted that the actual flow separates over the edge of the 1
nose creating a small re-circulation region close to the tip of Figure 3. The grid topology used in Euler the probe, which has been captured by the Navier-Stokes simCFD simulations ulation. As can be seen form Figure 2(b), the extend of the flow separation region in the streamwise direction is small, approximately 8% of the total cone surface length (L). Figure 4 shows the pressure distributions on the cone surface for inviscid and viscous flow solutions. The difference between the pressure distributions can be observed close to the nose where the re-circulation region exists. However, the inviscid pressure distribution closely follows the viscous solution for the most of the cone surface starting from xs /L ≈ 0.2. Due to the small difference between viscous and inviscid pressure distributions in the constant pressure region of the probe where the static ports are located, Euler solutions are used in this stochastic CFD study for computational efficiency.
p / pref
€ Navier‐Stokes CFD Euler CFD
x s /L
€
x s /L Figure 4. The static pressure distributions on the truncated cone surface obtained with Euler and NavierStokes simualtions for the mean probe geometry
€
6 of 13
M1
M1
M1
€
€
€
DN (inches)
δc (deg.) (a) Effect of δc on M1
(b) Effect of DN on M1
x c (inches) (c) Effect of xc on M1
Figure Carlo with 1000 samples. € 1 ) obtained with Latin Hypercube Monte € €5. Scatter plots of Mach number (M
IV. A.
Uncertainty Quantification Results and Discussion
Latin Hypercube Monte Carlo Results
In addition to the Point-Collocation NIPC method, Latin Hypercube Monte Carlo technique has been used to calculate various statistics for the wall static pressure p˜w at xs /L = 0.06 (xs is the distance from the tip on the cone surface and L is the total cone surface length), the cone-static pressure p˜c , and the Mach number M1 (˜ p indicates that the pressure p is scaled with the free-stream value). These output quantities were calculated at 1000 Latin Hypercube points (i.e., 1000 deterministic CFD solutions) each consisting of a cone angle (δc ), nose diameter (DN ) and a cone static pressure hole location (xc ) randomly sampled from the associated uniform probability distributions. Figure 5 shows the scatter of the Mach number obtained with the Monte Carlo simulations and gives a qualitative description of its dependence on the three uncertain input parameters (δc , DN , and xc ). In Figure 5(a), at a constant δc value, the scatter in M1 is due to the uncertainty in the other two parameters, DN and xc . Similarly, in Figure 5(b), the variation in M1 is due to the uncertainty in δc and xc for a fixed value of DN , and due to the uncertainty in δc and DN at a fixed value of xc in Figure 5(c). Examining these three figures, one can see that the uncertainty in the cone angle has more influence on the variation of the Mach number, since the vertical range of the scatter at fixed input parameter values in Figures 5(b) and 5(c) are much larger than those seen in Figure 5(a). This observation can also be interpreted as a qualitative sensitivity analysis for M1 using Monte Carlo data. A clear pattern for the dependence of M1 on the cone angle can be seen from Figure 5(a). As the cone angle increases, the Mach number decreases continuously. One can see a slight increase in the Mach number as the nose diameter is increased (Figure 5(c)). No particular dependency of the Mach number on xc can be observed by analyzing Figure 5(c). Table 1 gives the Latin Hypercube Monte Carlo statistics for the Mach number. The mean value of the Mach number is approximately 1.95. Compared to the nominal value of the free-stream Mach number (M1 = 2.0), there is a 2.5% bias difference in the calculated Mach number which may be due to neglecting viscous effects in the CFD simulations, the experimental calibration uncertainty for obtaining the Krasnov
Table 1. The Latin Hypercube Monte Carlo and Point-Collocation NIPC (p = 2) statistics for the Mach Number M1 . The 95% confidence intervals for the MC statistics were calculated using the Bootstrap method.
Statistics
MC
95% Confidence Interval of MC
Point-Collocation NIPC (p = 2)
Mean Standard Deviation Coefficient of Variation
1.95516 0.014360 0.0073448
[1.95427, 1.95602] [0.013897, 0.014809] [0.0071139, 0.0075787]
1.95513 0.014691 0.0075138
7 of 13
Table 2. The number of deterministic CFD solutions required for the evaluation of the Latin Hypercube Monte Carlo simulations and the Point-Collocation NIPC for the stochastic CFD study of the supersonic pressure probe. (p is the degree of the polynomial chaos expansion).
Monte Carlo
p=1
p=2
NIPC p=3
1000
8
20
40
number of CFD runs
p=4
p
70
112
curve parameters, and the numerical errors (discretization and iterative convergence error of CFD solutions). However the relative variation in the Mach number due to the specified geometric uncertainty is approximately 0.7%. as indicated by the coefficient of variation data. Table 1 also gives the 95% confidence interval for each Monte Carlo statistics, which was obtained with the Bootstrap Method. Calculating the confidence intervals provides a proper way of comparing the Monte Carlo statistics to the ones obtained with the polynomial chaos approach, especially when the number of Monte Carlo samples is limited. The advantage of the Bootstrap Method is that it is not restricted to a specific distribution, e.g. a Gaussian. It is easy and efficient to implement, and can be completely automated to any estimator, such as the mean or the variance. In practice, one takes at least 100 bootstrap samples to obtain a standard error estimate. In our computations, we used 500 bootstrap samples each consisting of 1000 observations selected randomly from the original Monte Carlo simulations by giving equal probability (1/1000) to each observation. B.
Point-Collocation NIPC Results
The Polynomial Chaos approach has been used to calculate various statistics of the output quantities of interest including the Mach number M1 . The coefficients in the polynomial expansions were calculated with the Point-Collocation NIPC method using Latin Hypercube samples (with np = 2) as the collocation points. The chaos expansions were obtained up to an order of five. Table 2 gives the computational cost associated with the Latin Hypercube Monte Carlo and the Point-Collocation NIPC approach. As can be seen from this table, even for the case with the highest order of polynomial chaos, Monte Carlo approach is nine times more expensive. Figure 6 shows the mean and coefficient of variation change for M1 as the polynomial order is increased. The Monte Carlo results are also included in the same figure. For all polynomial orders, the statistics obtained with the Point-Collocation NIPC approach falls within the confidence interval of the Monte Carlo results, which indicates that the level of Monte Carlo accuracy in the statistics can be achieved with all Mean
CoV
Mean
MC
95%
CI
CoV
MC
95%
CI
Mean
PC
CoV
PC
Mean
MC
CoV
MC
PC
Order
PC
Order
(a) Mean value of M1
(b) Coefficient of variation of M1
Figure 6. Mean and coefficient of variation (CoV) of Mach number (M1 ) obtained using Latin Hypercube Monte Carlo and Point-Collocation NIPC.
8 of 13
CDF( p˜ w )
€ PC
order
1
PC
order
2
PC
order
3
PC
order
4
PC
order
5
MC
p˜ w
p˜ w
(a) CDF of p˜w at xs /L = 0.06
(b) Histogram of of p˜w at xs /L = 0.06
€
€
CDF( p˜ c )
€ PC
order
1
PC
order
2
PC
order
3
PC
order
4
PC
order
5
MC
p˜ c
p˜ c
(c) CDF of p˜c
(d) Histogram of p˜c
€
€
CDF(M1 )
€ PC
order
1
PC
order
2
PC
order
3
PC
order
4
PC
order
5
MC
M1
M1 (e) CDF of M1
(f) Histogram of M1
Figure 7. Cumulative distribution function CDF and histogram for the static pressure (˜ pw ) at xs /L = 0.06, the € € cone pressure (˜ pc ) measured at xc , and the Mach number (M1 ).
9 of 13
CoV ( p˜ w ) MC
€
PC
order2
x s /L Figure 8. Coefficient of variation (CoV ) distribution for the static pressure on the cone surface obtained with Point-Collocation NIPC of order 2 and Latin Hypercube Monte Carlo
€ polynomial orders but at a much reduced computational cost. Table 1 gives the polynomial chaos statistics for the second order expansion. Similar to the Monte Carlo results, the mean Mach number is approximately 1.95 and the relative variation is 0.7%. In addition to the Mach number, the statistics and the uncertainty information for the other output quantities were also obtained with the non-intrusive polynomial chaos approach. In Figure 7, the cumulative distribution functions (CDF ) for the output variables p˜w at xs /L = 0.06, p˜c , and M1 are presented for all polynomial orders. The Monte Carlo results are also included in the same plots for comparison. As can be seen from these figures, the Point-Collocation NIPC results are very close to the CDF shapes obtained with the Monte Carlo approach for all variables. For the p˜w variable, the CDF obtained with the NIPC method perfectly matches with the CDF of the Monte Carlo starting from the second order expansion. For the other variables, (˜ pc and M1 ) the perfect match of the cumulative distribution functions can be seen even for the first order polynomial chaos approximations. The static pressure measured on the cone surface at xs /L = 0.06 (˜ pw ) has a larger variance compared to the cone pressure p˜c , which may explain why a second order polynomial expansion is required to get the best match in CDF. Although the measurement location of p˜c is uncertain (xc ), the range of the measurement location is at a region on the cone surface (0.69 ≤ xs /L ≤ 0.77) where the change in static pressure is expected to be much smaller. This difference in the variance can also be seen from the histogram plots given in Figures 7(b) and 7(d). It should be noted that although the variance of p˜w is larger, it does not depend on the uncertainty of input variable xc (the location of the cone pressure measurement). Therefore, the uncertainty in p˜w should be due to the variation in the cone angle and the nose diameter. This is also true for the variation of the static pressure distribution on the cone surface, which is shown in Figure 8. From this figure it can be seen that, the relative uncertainty indicated by the coefficient of variation can be as large as 15% for the static pressure values measured very close to the nose. As the measurement location is shifted downstream along the cone surface, the variation in the wall static pressure due to geometric uncertainty decreases and reaches a constant value of approximately 1% starting from xs /L = 0.2. These observations indicate that the uncertainty in the cone pressure measurement remains constant and minimum as long as the static pressure holes are located beyond 20% of the truncated cone surface length, which implies that a smaller-size probe design can be achieved by moving the pressure ports further upstream on the cone surface since the static ports are located at 73% of the overall cone surface length in the current probe configuration. In fact, beyond this location the mean value of the wall static pressure approaches to a constant value as shown in Figure 9. In the same figure, the uncertainty bars which include the pressure values within the 95% confidence interval are also shown. As expected, the uncertainty bounds are significantly larger close to the nose where xs /L ≤ 0.2. This plot also shows that the predictions with the second order polynomial chaos 10 of 13
p˜ w
p˜ w PC
order
2
MC
€
€
x s /L
x s /L
(a) Point Collocation NIPC of order 2
(b) Latin Hypercube MC
€
Figure 9. Mean static pressure distribution on the cone surface (˜ pw ).€ Each uncertainty bar includes the pressure values within 95% confidence interval.
expansion match perfectly with the Monte Carlo results. Using the polynomial chaos expansions, one can also calculate the sensitivity of the Mach number M1 to geometric input variables using ∂M1 ∂M1 ∂ξ1 = , (7) ¯ ∂ξ1 ∂ δ¯c ∂ δc ∂M1 ∂M1 ∂ξ2 = ¯ ¯N , ∂ξ2 ∂ D ∂ DN
(8)
∂M1 ∂M1 ∂ξ3 = . (9) ∂x ¯c ∂ξ3 ∂ x ¯c ¯ N = DN /(DN )mean , and Here, all input variables are scaled with their mean values (δ¯c = δc /(δc )mean , D x ¯c = xc /(xc )mean ) to obtain the scaled sensitivity information for the Mach number. Figure 10 shows the scaled sensitivity of the Mach number to the cone angle, nose diameter, and the cone static pressure location calculated from the second order polynomial chaos expansion of the Mach number. In this figure, the sensitivities are obtained for each parameter at their defined range while keeping the other parameters fixed at their mean values. As can be seen from the plots in Figure 10, the Mach number is an order of magnitude more sensitive to the variation in the cone angle than the variation in the other parameters. For example, at the mean values of the geometric parameters, the scaled sensitivities of Mach number to the cone angle, nose diameter, and the static pressure are -0.477043, 0.0374385, and -0.0439502 respectively. It should be also noted that the sensitivity information obtained with the polynomial chaos approach is consistent with the qualitative sensitivity information observed from the Monte Carlo results. ∂M1 /∂DN
∂M1 /∂δc DN = 1.0 x c = 1.0
€
∂M1 /∂ x c
δc = 1.0 x c = 1.0
€
DN = 1.0
€
€
(a) Sensitivity of M1 to δ¯c
€
DN
δc
€
δc = 1.0 €
¯N (b) Sensitivity of M1 to D €
xc
(c) Sensitivity of M1 to x ¯c €
Figure 10. The scaled sensitivity information for the Mach number M1 calculated using second order polynomial chaos expansions. All quantities are scaled with their mean values.
11 of 13
V.
Conclusions
The Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) Method has been applied to the stochastic CFD analysis of a pressure probe designed for three-dimensional supersonic flow measurements with moderate swirl. The supersonic pressure probe is in the shape of a truncated cone with a flat nose opening for total pressure measurement and includes four holes on the cone surface equally spaced in the circumferential direction for static pressure measurements. The simultaneous measurement of the total and the static pressure measurements enable the calculation of the Mach number at the probe location using the Krasnov’s similarity law. Due to the manufacturing tolerances, the probe can have geometric uncertainty which may effect the accuracy of the pressure measurements and the Mach number prediction. One of the objectives of the present study is to quantify the uncertainty in the pressure measurements and the Mach number due to the uncertainty in three geometric parameters: the cone angle (δc ), the nose diameter (DN ), and the location of the static holes measured on the cone surface from the nose (xc ). Each of these parameters were modeled as uniform random variables with a specified range and mean value based on the tolerances supplied by the manufacturer. The present stochastic CFD study is also performed to provide information for further refinement of the supersonic pressure probe design in the presence of geometric uncertainties. A full Navier-Stokes simulation performed with the mean probe geometry revealed that the inviscid pressure distribution closely follows the viscous solution for the most part of cone surface (starting from xs /L ≈ 0.2) except close to the truncated nose where a small flow separation region exists. Due to the small difference between viscous and inviscid pressure distributions in the constant pressure region of the probe where the static ports are located, the inviscid solutions are used in this stochastic CFD study for computational efficiency. The propagation of the geometric uncertainty to the output quantities of interest was performed with two methods: Point-Collocation NIPC method and Latin Hypercube Monte Carlo simulations. Both methods utilized deterministic Euler CFD solutions run at the selected Latin Hypercube sample points for a freestream Mach number of 2.0 at zero angle of attack. The uncertainty information obtained for various parameters with both methods were very close and all statistics obtained with a second order polynomial chaos expansion fell within the confidence interval of the Monte Carlo statistics. The second order NIPC required only 20 CFD solutions to obtain the uncertainty information, whereas the Monte Carlo simulations were performed with 1000 samples (CFD solutions), indicating the computational efficiency of the polynomial chaos approach. The mean value of the Mach number was obtained as 1.95. Compared to the nominal value of the free-stream Mach number (M1 = 2.0), there is a 2.5% bias difference in the calculated Mach number which may be due to neglecting viscous effects in the CFD simulations, the experimental calibration uncertainty for obtaining the Krasnov curve parameters, and the numerical errors (discretization and iterative convergence error of CFD solutions). However the relative variation in the Mach number due to the specified geometric uncertainty was approximately 0.7%. The stochastic results showed that the uncertainty in the cone pressure measurement remains constant and minimum as long as the static pressure holes are located beyond 20% of the truncated cone surface length. This information implies that a smaller-size probe design can be achieved by moving the pressure ports further upstream on the cone surface since the static ports are located at 73% of the overall cone surface length in the current probe configuration. The relative variation in the cone surface pressure can be as large 15% close to the nose location. The sensitivity analysis obtained from the polynomial chaos expansions reveal that the Mach number is an order of magnitude more sensitive to the variation in the cone angle than the uncertainty in the other geometric variables. Overall, this paper demonstrated efficient uncertainty quantification with Point-Collocation NIPC and CFD for the design and operation of an advanced aerospace measurement device which included geometric uncertainty due to manufacturing tolerances.
References 1 Hosder, S., Walters, R. W., and Perez, R., “A Non-Intrusive Polynomial Chaos Method For Uncertainty Propagation in CFD Simulations, AIAA-Paper 2006-891,” 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January, 2006, CD-ROM. 2 Hosder, S., Walters, R. W., and Balch, M., “Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Input Uncertain Variables, AIAA-Paper 2007-1939,” 9th AIAA Non-Dterministic Approaches Conference, Honolulu, HI, April, 2007, CD-ROM.
12 of 13
3 Walters, R. W. and Huyse, L., “Uncertainty Analysis for Fluid Mechanics with Applications,” Tech. rep., ICASE 2002-1, NASA/CR-2002-211449, NASA Langley Research Center, Hampton, VA, 2002. 4 Ghanem, R. and Spanos, P. D., “Polynomial Chaos in Stochastic Finite Elements,” Journal of Applied Mechanics, Vol. 57, March 1990, pp. 197–202. 5 Ghanem, R., “Stochastic Finite Elements with Multiple Random Non-Gaussian Properties,” Journal of Engineering Mechanics, January 1999, pp. 26–40. 6 Ghanem, R. G., “Ingredients for a General Purpose Stochastic Finite Element Formulation,” Computational Methods in Applied Mechanical Engineering, Vol. 168, 1999, pp. 19–34. 7 L. Mathelin, M.Y. Hussaini, T. Z. and Bataille, F., “Uncertainty Propagation for Turbulent, Compressible Nozzle Flow Using Stochastic Methods,” AIAA Journal, Vol. 42, No. 8, August 2004, pp. 1669–1676. 8 Xiu, D. and Karniadakis, G. E., “Modeling Uncertainty in Flow Simulations via Generalized Polynomial Chaos,” Journal of Computational Physics, Vol. 187, No. 1, May 2003, pp. 137–167. 9 Wiener, N., “The Homogeneous Chaos,” American Journal of Mathematics, Vol. 60, No. 4, 1938, pp. 897–936. 10 Walters, R., “Towards stochastic fluid mechanics via Polynomial Choas-invited, AIAA-Paper 2003-0413,” 41st AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January, 2003, CD-ROM. 11 Najm, H. N., “Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics,” Annual Review of Fluid Mechanics, Vol. 41, 2009, pp. 35–52. 12 Loeven, G. J. A., Witteveen, J. A. S., and Bijl, H., “Probabilistic Collocation: An Efficient Non-Intrusive Approach for Arbitrarily Distributed Parametric Uncertainties, AIAA-Paper 2007-317,” 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January, 2007, CD-ROM. 13 Debusschere, B. J., Najm, H. N., Pebay, P. P., Knio, O. M., Ghanem, R. G., and Maitre, O. P. L., “Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes,” SIAM Journal on Scientific Computing, Vol. 26, No. 2, 2004, pp. 698–719. 14 Reagan, M., Najm, H. N., Ghanem, R. G., and Knio, O. M., “Uncertainty Quantification in Reacting Flow Simulations through Non-Intrusive Spectral Projection,” Combustion and Flame, Vol. 132, 2003, pp. 545–555. 15 Isukapalli, S. S., “Uncertainty Analysis of Transport-Transformation Models, PhD Dissertation,” Tech. rep., Rutgers, The State University of New Jersey, New Braunswick, NJ, 1999. 16 L. Mathelin, M.Y. Hussaini, T. Z., “Stochastic Approaches to Uncertainty Quantification in CFD Simulations,” Numerical Algorithms, Vol. 38, No. 1, March 2005, pp. 209–236. 17 Eldred, M. S. and Burkardt, J., “Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification, AIAA Paper 2009-0976,” 47th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL, January, 2009. 18 Wiener, N. and Wintner, A., “The Discrete Chaos,” American Journal of Mathematics, Vol. 65, No. 2, 1943, pp. 279–298. 19 Krasnov, N. F., Aerodynamics of Bodies of Revolution, American Elsevier Publishing Company, Inc., New York, New York, 1970. 20 Maddalena, L., Hosder, S., Bonanos, A., and Dimotakis, P., “Extended Conical Flow Theory for Design of Pressure Probes in Supersonic Flows with Moderate Flow Angularity and Swirl, AIAA-Paper 2009-1072,” 47th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL, January, 2009. 21 GASP User Manual, AeroSoft, Inc., Blacksburg, Virginia, 1997.
13 of 13
