Surrogate-Based Modeling of Cryogenic Turbulent Cavitating Flows

Transcription

1 Proceedings of the 7 th Internationa Symposium on Caitation CAV009 Paper No. 77 August 17-, 009, Ann Arbor, Michigan, SA Surrogate-Based Modeing of Cryogenic Turbuent Caitating Fows Abstract The cryogenic caitation has critica impications on the performance and safety of iquid rocket engines. In this study, a systematic inestigation based on the surrogate modeing techniques is conducted to assess and improe the performance of a transport-based cryogenic caitation mode. Based on the surrogate mode, goba sensitiity anaysis is be conducted to assess the roe of mode parameters reguating the condensation and eaporation rates, and uncertainties in materia properties, specificay, the apor density and atent heat. The surrogate modes considered incude the response surface, radia basis neura network, Kriging, and a weighted aerage composite mode combining a surrogates. It is reeaed that the apor density and the mode parameter controing the eaporation rate are more critica than atent heat and the mode parameter controing the condensation rate. Based on the recommended mode parameter aues, better prediction of the cryogenic turbuent caitation can be attained. 1. Introduction For the fue deiery of iquid rockets, caitation occurs as the oca pressure is ower than apor pressure, which can seriousy compromise the engine performance and structura integrity [1,]. At cryogenic conditions, the iquid/apor density ratio is typicay ower than that under non-cryogenic conditions. Thus, to sustain a comparabe caity size, heat transfer caused by atent heat becomes more important due to higher mass transfer rates. Consequenty, the iquid surrounding the apor caity has more significant eaporatie cooing in the cryogenic iquids. In addition, other transport properties such as therma conductiity and thermodynamic properties such as apor pressure of cryogenic fuid are ery sensitie to temperature ariations. Substance Chien-Chou Tseng Department of Mechanica Engineering niersity of Michigan, Ann Arbor, MI, SA Heat capacity (J/Kg K) Density ratio Therma conductiity (W/mK) Latent heat (KJ/Kg) H O(98k) N (83k) H (0k) Tabe 1. Variation of physica properties for water (98k), iquid nitrogen (83k), and iquid hydrogen (k) on saturation cures. [5] Wei Shyy Department of Aerospace Engineering niersity of Michigan, Ann Arbor, MI, SA (a)vapor pressure s. temperature (b)liquid density s. temperature (c)density ratio (iq./apor) s. temperature (Soid ines represnt for water and use bottom and eft as x-axis and y-axis in each figure; dash ines represnt for iquid nitrogen and use top and right as x- axis and y-axis in each figure) Figure 1. Variation of physica properties for iquid nitrogen and water aong saturation cure. [5] Therefore, for cryogenic fuids, the energy transport equation needs to be incuded in the caitation mode [3,4]. Representatie aues of these quantities and the pressuretemperature saturation cures are summarized in Tabe 1 and Figure 1 [5]. The dynamic simiarity for isotherma case, such as water is goerned by the caitation number [1,,3]: σ = ( P P )0.5ρ (1) Where the caitation number σ is based on a constant apor pressure P at inet temperature T. P, ρ, and are reference pressure, iquid phase density and free stream eocity respectiey. For cryogenic caitation, the actua oca caitation number σ needs to be corrected according to the oca temperature T: σ = ( P P ( T )) / 0.5ρ () By the foowing first order approximation: 1 dp ρ ( σ σ ) = ( T T ) dt (3) dp T σ = σ ; T = T T < 0 dt 0.5ρ Equation (3) ceary shows the temperature dependency of caitation, and the oca temperature drop in cryogenic caitation wi produce a noticeabe rise for the oca caitation 1

2 number σ and hence suppress the caitation intensity [3,4]. The detai impact for the therma-sensibe materia properties to caitation mode wi be introduced ater. The numerica modeing of caitation argey foow two main categories: interface tracking methods with indiidua phases separatey treated [6,7], and homogeneous fow modes based on a singe-fuid framework with fuid properties estimated based on the iquid-apor mixture ratios [3,4,6,7,8,9,10,11,1,13,14,15]. Differences between the arious modes in the second category mosty come from the reation that defines the density fied. For oeriew of the arious modeing approaches, see, e.g., [3,4,8]. A homogeneous fow mode utiizing the framework of the transport-based equation (TEM) is adopted in the present study. In this method, the information of the apor oume/mass fraction distribution is obtained in a modeed transport equation based on the mass transfer between apor and iquid phases. This approach is we documented, see, e.g., [3,4,8]. To study the impact of the cryogenic mode parameters and fuid properties on the predictions, we conduct a mode improement exercise utiizing a goba sensitiity anaysis. We use the method deeoped by Sobo [16]. This method aows decomposition of a suitabe measure of prediction into the components of indiidua ariabes from which we can easiy cacuate the impact of each ariabe. To faciitate the framework which heps us probe the goba sensitiity of the caitation mode and fuid uncertainties, we wi first construct suitabe surrogate modes [17]. The practica utiity of surrogate modeing for design, optimization and sensitiity anaysis is we estabished [18,19]. There are many surrogate modes, but the mode that represents a particuar function the best is not known in adance. Thenthe predictions using different surrogate modes hae a certain amount of uncertainty. Goe et a [4,19] suggested that combinations of mutipe surrogate modes can be beneficia to quantify and to reduce uncertainties in predictions. They proposed a PRESS-based weighted aerage, namey PWS, (PRESS is the predicted residua sum of square) to reduce the mode uncertainties. Since the fideity of surrogate modes is critica in determining the success of the sensitiity anaysis, we wi use different surrogate modes to hep ascertain the performance measures. In this study, we use four surrogate modes, poynomia response surface approximation (PRS, [0]), Kriging (KRG, [1]), radia basis neura network (RBNN, []) and PRESS-based weighted aerage (PWS) surrogate mode constructed by using the preious three surrogates [4,17,19]. These surrogate modes are used to caibrate the mode parameters of the present transport based caitation mode for cryogenic caitation. The surrogate-based goba sensitiity anaysis can hep us inestigate the uncertainties from fuid properties and then identify the optima mode parameters to improe the prediction performance and robustness of cryogenic caitation modes. For turbuence cosure, the ensembe-aeraged modeing with a two-equation cosure [3] aong with a fiter-based mode (FBM) [4] is utiized. The approach reduces the infuence of the turbuent eddy iscosity based on the oca numerica resoution, essentiay bending direct numerica simuation (DNS) and conentiona turbuence mode in a singe framework. Specificay, the ee of the turbuent iscosity is corrected by comparing the turbuence ength scae computed from the turbuence cosure and the fiter size based on the oca mesh spacing. This approach can be aso categorized as Very Large Eddy Simuation (VLES) [5,6]. As documented in a preious study [7], the inet conditions of the turbuence mode can criticay affect the outcome of caitation structure. The fiter-based mode can hep significanty reduce the uncertainty in this regard. In this study, we re-examine the surrogate-based cryogenic modeing efforts preiousy reported by Goe et a. [4] to assess the predictie capabiity of turbuent caitating fows.. Goerning Equations and Numerica Approaches The set of goerning equations for cryogenic caitation under the homogeneous-fuid modeing consists of the conseratie form of the Fare-aeraged Naier-Stokes equations, the enthapy-based energy equation (for cryogenic caitation), the k-ε two-equation turbuence cosure, and a transport equation for the iquid oume fraction [3,4,8]. The continuity, momentum, enthapy, and caitation mode equations are gien beow. A computations presented beow are based on the steady-state equations. ( ρ u ) m = 0 x ( ρ uu ) p u u u m i i k = + [( µ + µ )( + δ )] L T i x x x x x 3 x i i k µ µ h m T [ ρ u ( h+ f L)] = [( + ) ] m x x Pr Pr x m T α + ( u ) = m& + m& (7) x where ρ m is the mixture density, u denotes the components of eocity, p is the pressure, µ and µ t are the mixture aminar and turbuent iscosities, respectiey, h is the sensibe enthapy, f is the apor mass fraction, L is the atent heat of aporization, Pr is the Prandt number, α is the iuid oume fraction, and m + and m are the source and sink terms for the caitation mode. The subscript t denotes turbuent properties, represents the iquid state, represents the apor state, and m denotes the mixture properties. The mixture property φ m and the apor mass fraction are, respectiey, expressed as φ = φα + φ ( 1 α ) (8) f m ρ ( 1 α ) = (9) ρ m The temperature can be interpoated based on enthapy in the data base [5]. We negect the effects of kinetic energy and iscous dissipation terms in Equation (4) (O(Re -0.5 ), Re is around 10 6 ) because the temperature fied is mainy contributed by the eaporatie cooing in cryogenic caitation. (4) (5) (6)

3 .1.Transport-Based Caitation Mode The source term m + and sink term m in Equation (7) represent for condensation and eaporation rates. They hae been deried from arious aspects, incuding dimensiona argument with empirica support [3,4,8,10,11,1], force baance based on the interfacia dynamics [3,8,9], and estimate of the bubbe growth rate through the Rayiegh-Pesset equation [13,14,15]. Numericay, [3,4,8,9,11,1] utiized pressure-based methods, and [10,1,16,13,14,15] empoyed the density-based methods. The iquid-apor eaporation and condensation rates for the present transport-based caitation mode [3,4,8,10,11,1] are respectiey shown as foowing: C α ρ min(0, p p ) dest m& = t ρ ρ (0.5 ) C (1 α ) max(0, p p ) + prod m& = (10) t (0.5 ρ ) where and are the empirica constants, is the reference eocity scae, and t is the reference time scae, which is the characteristic ength scae D diided by the reference eocity scae (t =D/ ). For non-cryogenic fuids ike water, the constants are specified =1 and =80 [3,8]. As for iquid nitrogen, the constants are chosen as =0.68 and =54.4, and for iquid hydrogen, =0.8 and =54.4 are suggested [3]. Further modifications are conducted to get a better prediction with =0.639 and =54.4 for iquid nitrogen, and =0.767 and =54.4 for iquid hydrogen [4]. For cryogenic caitation simuations, the temperature dependent properties are updated from a comprehensie data base [5] throughout the course of computations for eery iteration... Thermodynamics Effects The impact of therma effects in cryogenic caitation due to phase change on temperature prediction has been aready shown in Figure 1. These thermo-sensibe materia properties wi affect the energy equation in Equation (6) and caitation sink/source terms in Equation (10). First, the atent heat L in Equation (6) appears as a noninear source term and represents the atent heat transfer rate during the phase change. The spatia ariation of the thermodynamic properties together with the eaporatie cooing effect is embedded into this equation and causes a couping with the set of goerning equations [3,4]. As for the caitation sink/source terms in Equation (10), we can assess the impacts due to the thermo-sensibe materia properties by using Tayor s series and negect the higher order terms. We first consider the sink term m as the pressure is smaer than the apor pressure based on the oca temperature [4,7], or in other words, pressure coefficient C p is smaer than σ. Furthermore, the minus sign here in Equation (11) is for conenience to show a arger eaporation strength wi correspond to a arger positie aue of m : C α ρ min(0, p p ( T)) dest m& = ( ) = βr( T) min(0, C + σ) p t ρ 0.5ρ dr dp T = β( R( T ) + T+...)( C + σ +...) p dt dt 0.5ρ T dr dp T = β( R( T )( C + σ ) + ( C + σ ) T R( T ) +...) p p dt dt 0.5ρ T T T (11) where β is α /t and R is the temperature-dependent iquid/apor density ratio, Simiary for source term m + as the pressure is arger than the apor pressure (C p is arger than σ): C (1 α ) prod max(0, p p ( T)) + m& = = γ max(0, C + σ ) p t 0.5ρ dp T = γ (( C + σ ) +...) p dt 0.5ρ T (1) Where γ is (1-α )/t. Pease note that T=T-T <0 in both Equation (11) and (1), which is aso defined in Equation (3) as eaporatie cooing occurs. It can be concuded that the competing infuence of the therma effects in the caitation mode comes from two ways from Equation (11) as eaporatie cooing occurs: (1) therma rate of change of iquid/apor density ratio dr/dt which is negatie in Figure 1(c) together with C p +σ <0 and T<0 as eaporation occurs, wi tend to enhance the strength of m and () therma rate of change of apor pressure dp /dt which is positie in Figure 1 to suppress m. It is aso obious that the impacts of therma effects wi change significanty for different working temperature and pressure due to the non-inear ariation of materia properties from energy equation in Equation (6) and caitation sink/source terms in Equation (10)..3. Turbuence Mode The k-ε two-equation turbuence mode with a wa function treatment is presented as foows [3]: ( ρ u k) µ k m T = P ρ ε + [( µ + ) ] t m L x x σ x k ( ρ uε ) m ε ε µ ε T = C P C ρ + [( µ + ) ] ε t ε m L 1 x k k x σ x ε (13) (14) where the production term of turbuent kinetic energy (P t ) and the Reynods stress tensor are defined as: ui P = τ ; τ = ρ u u t i i m i x ρ kδ u u m i i ρ u u = µ ( + ) m i T 3 x x i (15) with C ε1 =1.44, C ε =1.9, σ ε =1.3,σ k =1.0. The turbuent eddy iscosity is defined as: ρ C k m µ µ =, C = 0.09 (16) T µ ε 3

4 As mentioned aboe, a fiter-based mode (FBM) [4] is aso adopted. This mode imits the infuence of the eddy iscosity based on the oca numerica resoution, essentiay forming a combined direct numerica simuation and RANS mode. Specificay, the ee of the turbuent iscosity is corrected by comparing the turbuence ength scae and the fiter size, which is based on the oca meshing spacing: k ε µ = 0.09ρ min(1, ) T m ε 3/ k (17) By imposing the fiter, the turbuence ength scae wi not be resoed if it is smaer than the fiter size. The fiter size is chosen to be comparabe to the maximum grid size: = max(, ) (18) present grid Thus if the grid resoution is significanty smaer than the turbuence ength scae in the entire fow fied, the soution wi approach that of a direct numerica simuation; for inadequatey resoed computations, the RANS mode is recoered. Simiar concepts can be found in studies of VLES [5,6]..4.Numerica method Detaied numerica procedures for the caitation mode and associated fuid dynamics equations adopted here utiize a modified pressure-based approach for arge density ump as we as therma effects, as reported in [8,9] The controed ariation scheme (CVS) [8] is appied to discretize the conection scheme, and centra difference is used for both pressure and diffusion terms. The CVS scheme can preent the osciations under sharp gradients caused by the phase change whie presering second-order accuracy esewhere. As for the boundary conditions, iquid oume fraction, eocity, temperature and turbuent quantities are specified at the inet. For the outet, pressure and other fow ariabes are extrapoated. On the was, pressure, iquid oume fraction, and turbuent quantities are extrapoated aong with no-sip and adiabatic conditions. Additionay, the pressure at the reference point (P ) in the upstream is aso fixed to define the caitation number σ [8,9]. Based on the eddy-to-aminar iscosity ratio at the inet, the inet turbuent quantities can be gien as foowing: 3 C ( ), µ k k = I ε = ν L ( µ T / µ L ) inet (19) where I is turbuence intensity (% for entire study), and eddyto-aminar iscosity ratio, is equa to1000 in a the cases during the current study. 3. Resuts and Discussion 3.1. Test Geometry We simuate fow oer a D quarter hydrofoi in Figure with the experimenta measurements by Hord [9]. Since we hae iustrated in Figure 1 and used Equation (11) and (1) to highight the impacts of therma effects wi change significanty for different temperatures, we seect both Case 90C and 96B to caibrate the roe of mode parameters and uncertainty of fuid properties. Figure and Tabe summarize the geometries, corresponding boundary conditions, and the fow conditions of the test cases seected to aid the mode aidation. Substance Case σ Re T Liquid nitrogen 90C K Liquid nitrogen 96B K Tabe. Summary of simuation setups and fuid properties Figure. Schematic of the geometries and the boundary conditions of the cases considered. 3.. Pressure and Temperature Predictions Figure 3 compares the predicted and experimentay measured pressure and temperature profies [9] on the hydrofoi surface with =0.68 and =54.4 suggested in [3]. Oera, the caitation and turbuence modes with fiter (FBM) can consistenty capture the main features of both pressure and therma profies. The temperature drop inside the caity in Figure 3(b) and (d) aso ceary demonstrates the eaporatie cooing resuting from cryogenic caitation. (a)pressure, 90C (c)pressure, 96B (b)temperature, 90C (d)temperature, 96B Figure 3. Pressure and temperature of cryogenic caitation aong surface. In Figure 4, we compare the present cryogenic mode soution with the isotherma soution for Case 90C, obtained by using the identica mode with =0.68 and =54.4 except that the energy equation is not inoked. Ceary, the therma fied does affect the caity structures. The caity size in Figure 4(b) is reduced due to the therma effect because the temperature drop inside the caity in Figure 4(a) decreases the oca apor pressure and hence increases the oca caitation number (pease refer to Equation (3)), resuting in a weaker 4

5 caitation intensity and higher oera iquid oume fraction in the caity (as shown in Figure 4(c)). Besides, the pressure inside the caity is steeper under the cryogenic condition than that under the isotherma condition in Figure 4(d) due to the ariation in therma rate of change of apor pressure dp /dt. The purpose of Figure 4 is to iustrate the therma effects coud significanty change the pressure fieds and caity structures under cryogenic conditions. (a) α with energy equation (b) α without energy equation (c) α aong surface (d)pressure aong surface Figure 4 Comparisons for Case 90C between resuts with/without energy equation 3.3 Surrogates-based Goba Sensitiity Assessment Since minor changes in fow enironment can ead to substantia changes in the predictions in cryogenic enironment, it is imperatie to appraise the roe of mode parameters and uncertainties in materia properties on the predictions. In this section, we characterize the parameters using surrogate-based goba sensitiity anaysis (GSA) and then caibrate the cryogenic caitation mode parameters. We use poynomia response surface approximation (PRS, [0]), Kriging (KRG, [1]), radia basis neura network (RBNN, []), and a weighted aerage surrogate mode (PWS) for approximation of response. [4,19]. We use ariance-based, non-parametric GSA method, proposed [16] to eauate the sensitiity of predictions with respect to mode parameters and materia properties. In this method, the obectie function is decomposed into additie functions of ariabes and interactions among ariabes. This aows the tota ariance (V) in the obectie function to be expressed as a combination of the main effect of each ariabe (V i ) and its interactions with other ariabes (V i ). The sensitiity of the obectie function with respect to any ariabe is measured by computing its sensitiity indices. The sensitiity indices of main effect (S i ) and tota effect ( S are gien as foows: tota i ) of a ariabe V i tota ( ), V i+ S V iz i = S V i = (0) V We choose,, ρ,, and L as design ariabes, whie hoding the Re and σ constant for the gien cases. The performance of predictions for the cryogenic caitation modes are represented by root mean square (RMS) aues of the differences between computed and experimenta aues aong hydrofoi surface for temperature ( ) and pressure ( ) as our obecties. The mode parameters, and, ary from to 0.68 and 46. to 54.4 respectiey. The materia properties ρ and L are perturbed within ±10% of the aue they assume from the NIST database [5]. These two empirica constants and directy contro the eaporation and condensation rate ia the caitation mode. Besides, ρ, as a fuid property which dominates the eaporatie cooing, aso appears directy in caitation sink term, and L wi determine the energy absorb or reease during the phase change. Therefore, these four mode parameters and fuid properties are seected as our design ariabes. To faciitate the deeopment of surrogate modes, 70 training points are seected using combined face-centered cubic composite design (FCCD, 5 points) and Latin hypercube samping (LHS, 45 points). Fie additiona test points which are not incuded in the 70 training points are used to aidate the surrogate modes for both case 90C and 96B. We eauate and for each data point using CFD simuations and construct PRS, Kriging, RBNN, and PWS modes of both obecties in normaized ariabe space. A ariabes and obecties are normaized such that 0 corresponds to the minimum aue and 1 corresponds to the maximum aue. Normaized ariabes and obecties are denoted by a superscript. We use second order poynomias for PRS and a spread coefficient=0.4~0.7 for RBNN. Reeant detais of the quaity of fit of surrogate modes are documented in [4]. The smaest root mean square of PRESS (PRESS is the predicted residua sum of square) in Appendix Tabe A1 indicates that the KRG mode has the best performance, whie RBNN has the worst oera performance. The contribution of different surrogate modes to the PWS mode is gien by the weights in Appendix Tabe A. Since the performance of KRG is the best, its weight is aso the greatest. There are additiona fie test points to aidate the surrogates. Appendix Tabe A3 shows the ocations of these points in the normaized design space for both Case 90C and 96B. The simuation resuts are presented to compare with the prediction of surrogates in Appendix Tabe A4 and A5 for Case 90C, and Appendix Tabe A6 and A7 for Case 96B. Due to the best performance in error estimate in Appendix Tabe A1 and tests for additiona fie sampes from Appendix Tabe A4 to A7, we wi use KRG to demonstrate the goba sensitiity anaysis. Appendix Tabe A8 and A9 show that the interactions among parameters are ery strong for pressure since the difference between main and tota sensitiity indices are obious for Case 90C and 96B. As for temperature, Case 96B has stronger interactions than those of Case 90C. From Figure 5 and 6 which eauate the weights of each ariabes ia goba sensitiity anaysis as pie-charts, and ρ, are ery important for, and whie and L don t hae noticeabe contribution in both cases. Besides, the weights of are ery simiar for Case 90C and 96B. As for, the 5

6 importance of L ceary increases, and een becomes the same important as and ρ in Case 96B, whie in Case 90C, the weight is ust around 6%. This indicates that the therma effects and importance of the thermodynamics property wi infuence the therma fied more significanty as temperature increases (Case 96B has higher inet temperature than Case 90C). ρ Figure 5. Pie-chart of goba sensitiity anaysis for Case 90C (KRG (Kriging), tota indices) ρ Figure 6. Pie-chart of goba sensitiity anaysis for Case 96B (KRG (Kriging), tota indices) is not important within this design space from the piechart, and this impies that the sensitiity of condensation term is not significant compared with eaporation term or. This shoud be because caitation initiates from eaporation, and this behaior aready decides how strong the caitation dynamics and how ow the iquid oume fraction wi be inside the caity. Besides, the condensation strength wi aso depend on the eaporation strength. Just imagine if we gie a ery ow strength on eaporation term, the condensation dynamics wi sti be weak een is arge. This is because the apor inside the caity wi not be sufficient for source term. Thus the weight of is much more important than in this design space. As we know, the caitation terms can be expressed as foowing in Equation (10): C α ρ min(0, p p ) dest m& = t ρ ρ L L (0.5 ) C (1 α ) max(0, p p ) + prod m& = t (0.5 ρ ) Therefore we can group /t /ρ and /t together to show the ariation in strength of caitation sink and source term under the combinations of design ariabes, and then ρ L L ρ normaize these aues in Figure 7 and 8. This is the direct impact which indicates the direct appearance of design ariabes in the caitation mode. A the normaized aues here are from the preious simuation resuts of the 70 training points. Figure 7 and 8 show the distributions of P diff s. ( /t /ρ ) are quite the same for both Case 90C and 96B. This is because both cases hae consistent pie-charts and weights in the pressure prediction in Figure 5 and 6. As, ( /t /ρ ) is sma in Figure 7 and 8, the sink term is not strong enough so that the caity size is too sma, and hence P diff wi be arge. When ( /t /ρ ) goes up to certain moderate aue, the corresponding eaporation term gies a more suitabe caity size with a smaer aue of P diff. For een arge aues of ( /t /ρ ), the caity sizes wi be too arge, and wi increase again. This ceary indicates that there exists a suitabe range for sink term or to obtain good pressure predictions. For the same aue of ( /t /ρ ), different P diff and T diff can sometimes be obtained. This is because different fuid properties, namey ρ or L, infuence the fow fieds. We haen t reay incuded this factor in Figure 7 and 8, and refer it as indirect impact. As for ( /t /ρ ) s. T diff, we sti can roughy see simiar trends as shown in ( /t /ρ ) s. P diff, but the distributions are not so consistent for both Case 90C in Figure 7 and 96B in Figure 8. Again, this is due to different pie-charts of for both cases as shown in Figure 5 and 6. This aso impies different impacts of therma effects at different temperatures which we hae aready stated before. For ρ, besides direct appearance in the sink term, it is aso part of fuid property. As a fuid property, it can change the aue of mixture density to exhibit different eaporatie cooing inside the caity, and hence affects the caity size and caitation dynamics. Therefore, it owns both direct and indirect impacts, and we can see its weight of from goba sensitiity anaysis is een arger than in Figure 5 and 6. As for ( /t ), there is reay no trend for P diff and. This is because from the pie-chart in Figure 5 and 6, we can see the importance of is insignificant, and a these distributions in Figure 7 and 8 woud be mainy due to the contributions of sink term. Therefore the random distributions for ( /t ) s. P diff and in Figure 7 and 8 hep us aidate the insignificant weight of in Figure 5 and 6 from another iewpoint. 3.4 Optimization of Mode Parameters In the preious section, we obsered that one of the mode parameters significanty infuences the performance of the present cryogenic caitation mode. Therefore we fix another mode parameter at 54.4 which is not reay infuentia on predictions, and assume the temperature-dependent materia properties ρ and L obtained from the NIST database [5] to be correct. Then we aow the mode parameter to ary between and 0.68, and try to find the optimization aues which gie and as ow as possibe. There are 11 training points within the design space with equa spacing. From the simuations shown in Figure 9, we can see for Case 90C, the trends of these two obecties are amost opposite, and for Case 96B, these two obecties are of the same trends. 6

7 . ( /t /ρ ) ( /t /ρ ) (a) 90C (b) 96B Figure 9. Location of points ( ) and corresponding responses ( is shown on eft y-axis, and is shown on right y-axis) used for caibration of the cryogenic caitation mode. Figure 7. Iustration of the direct impact of Case 90C. ( /t ) ( /t ) The function space and the ariabe space iustration of Pareto optima front (POF) obtained through different surrogate modes are shown in Figure 10 and 11. We obsere that different POFs obtained by using mutipe surrogate modes are cose to one another in both function and ariabe spaces (Besides PRS for in Case 90C). We note that the pressure fuctuations pay a more important roe in determining the caitation dynamics and the oadings on fuid machinery. Consequenty, more accurate pressure prediction is our primary obectie. We seect a soution on the POF for aidation, such that notabe reduction in can be reaized without incurring significant deterioration of.. Therefore the optima wi be 0.65 from Figure 10 and 11. It aso indicates that different therma effects wi not significanty affect the choice of optima. ( /t /ρ ) ( /t /ρ ) (a) Variabe space (b) Function space Figure 10. Pareto optima front and corresponding optima points for Case 90C. ( /t ) ( /t ) Figure 8. Iustration of the direct impact of Case 96B. To represent the responses and using surrogate modes, we sampe data using CFD simuations from Figure 9. and estimated by surrogate modes shown in Figure 10(a) and 11(a) ceary exhibit the same trend as simuations in Figure 9. As before, we construct PRS, KRG, RBNN, and PWS modes. The error estimates and the weights associated with different surrogates in PWS mode are summarized in Tabe A10 and A11. Besides PRS for in Case 90C, a the surrogates yied a good accuracy. (a) Variabe space (b) Function space Figure 11. Pareto optima front and corresponding optima points for Case 90C. (KRG,Kriging; PRS, Poynomia Response Surface; RBNN, Radio Basis Neura Network; PWS, PRESS-Based Weighted Aerage Surrogate.) 7

8 3.5 Optimization for Liquid hydrogen We hae repeated simiar mode caibration process for iquid nitrogen by considering Case 49D (σ =1.57, T =0.7K, Re = 10 7 ) and 55C (σ =1.49, T =.K, Re = ), the optima aue of wi be Improement by Fiter-based Turbuence Mode Compared with Goe. et a [4], our current resuts exhibit improement in both pressure and temperature prediction by utiizing fiter-based mode (FBM, [4]). This mode heps us reduce the uncertainties from inet turbuent quantities. The improement is summarized in Tabe 3. Pease note that our is 0.65 for iquid nitrogen and 0.78 for iquid hydrogen, whie Goe.et a use and respectiey. Case Fuid Current resuts Goe et a [4] Current resuts Goe et a [4] 90C Nitrogen B Nitrogen D Hydrogen C Hydrogen Tabe 3. Performance compasions between current study and Goe. et a [4]. We hae aso used =0.65 to other iquid nitrogen cases isted in the experiment [9], and the simuation resuts are aso ery consistent to experimenta data in Tabe 4. This exercise heps us to aidate our mode parameters are aso suitabe for other cases. Case σ T (K) 83B C A Tabe 4. =0.65 for other iquid nitrogen cases. 4. Concusions In this study, we choose,, ρ,, and L as design ariabes and use different surrogate modes, PRS, KRG, RBNN, and PWS, to construct the response of pressure prediction and temperature prediction for cryogenic caitation. As documented in our preious study [7], the inet conditions of the turbuence mode can criticay affect the outcome of caitation structure. The fiter-based mode can hep significanty reduce the uncertainty in this regard. We hae re-examined the surrogate-based cryogenic modeing efforts preiousy reported by Goe et a. [4] based on a more refined turbuence approach to better probe the combined turbuent caitating fow predictie capabiities. It is found that the performance of the current caitation mode is affected more by mode parameter associated with the eaporation term and the fuid property ρ which contros the eaporatie cooing. The condensation term ( ) is not important at a within this design space, and atent heat L becomes significant ony in temperature prediction. We hae aso obsered that there is a range of eaporation term or which can gie better accuracy of pressure prediction. We recommend =0.65 and 0.78 for iquid nitrogen and hydrogen respectiey, and is 54.4 for both fuids. 5. Acknowedgments The present efforts hae been supported by NASA Consteation niersity Institute Program (CIP). 6. Appendix Acronyms in this appendix: PRS Poynomia Response Surface [0] KRG Kriging [1] RBNN Radio Basis Neura Network [] PWS PRESS-Based Weighted Aerage Surrogate [4,17,19] PRESS The predicted residua sum of square R ad Coefficient of mutipe determination [0] (better fit as it is coser to 1) Surrogate 90C 96B 90C 96B PRESS rms of PRS 6.38% 11.90% 9.11% 10.0% R ad of PRS PRESS rms of KRG.97%.93%.48% 6.6% PRESS rms of RBNN 13.91% 11.67% 13.31% 19.03% PRESS rms of PWS 3.97% 5.44% 5.50% 9.0% Tabe A1 Error estimate for different cases and surrogates (70 training points) Surrogate 90C 96B 90C 96B PRS 44.% 7.5% 31.8% 36.4% KRG 47.4% 44.5% 45.0% 39.6% RBNN 8.4% 8.3% 3.% 4.0% Tabe A Weights associated with different surrogate modes (70 training points) No No No No No Tabe A3. Test points inside the normaized design space PRS KRG RBNN PWS CFD -.3% -.7% -8.0% -3.6% % -.7% -5.4% -5.0% % 3.0% 1.0% 14.8% % -5.3% -1.6% 4.3% % 0.% -3.7% -.7% Tabe A4. Predictions error of for case 90C ρ L 8

9 PRS KRG RBNN PWS CFD -0.% -0.5% 1.% 0.0% % -1.7% -9.1% -.4% % 1.8% -0.7%.7% % -1.1% 1.1% 0.8% % 0.1% -0.% -1.4% Tabe A5. Predictions error of for case 90C PRS KRG RBNN PWS CFD -15.9% -5.1% 1.%.8% % -3.6% 46.3% -0.9% % -6.6% -6.0% 0.4% % -6.8% 8.1% -4.6% % -4.6% 8.8% -.7%.6871 Tabe A6. Predictions error of for case 96B PRS KRG RBNN PWS CFD 0.6% 0.3% -.% 0.% % 6.4% 1.5% -4.9% % 6.7% 0.% -5.5% % 0.9% -1.6% -0.4% % 1.0% -1.3% 0.1% Tabe A7. Predictions error of for case 96B (The coumn for CFD in Appendix Tabe A4 to A7 denotes simuation resuts before normaized.) 0.176/ E-5/ / / /0.53 1E-4/ / /0.06 Tabe A8. Goba sensitiity for 90C with KRG (main/tota effect) 0.035/0.61 E-4/ / / / / / /0.404 Tabe A9. Goba sensitiity for 96B with KRG (main/tota effect) ρ ρ Surrogate 90C 96B 90C 96B PRESS rms of PRS 16.01% 4.46% 3.8% 6.00% R ad of PRS PRESS rms of KRG 7.01% 4.87% 1.89% 1.91% PRESS rms of RBNN 4.00% 6.74% 1.56% 0.73% PRESS rms of PWS 7.45% 5.10% 3.0% 1.83% Tabe A10 Error estimate for different cases and surrogates (11 training points) Surrogate 90C 96B 90C 96B PRS 0.8% 35.8% 3.6% 13.5% KRG 36.9% 34.5% 37.1% 39.5% RBNN 4.3% 9.6% 39.3% 47.0% Tabe A11 Weights associated with different surrogate modes (11 training points) L L 7. Nomencature σ Free stream caitation number σ Caitation number based on the oca temperature C ε1,c ε Coefficients of k-ε turbuence mode σ ε,σ k Coefficients of k-ε turbuence mode C p Pressure coefficient D Characteristic ength scae f Vapor mass fraction h Enthapy I Turbuence intensity K Turbuent kinetic energy L Latent heat m +, m - Source and sink terms in the caitation mode L norm between experiment and predicted pressure Pr Prandt number Pt Production term of turbuent kinetic energy P Saturation apor pressure Re Reynods number S Sensitiity indices T Temperature L norm between experiment and predicted temperature t Reference time scae, t =D/ Reference eocity u Veocity x Space ariabe V tota ariance α Liquid oume fraction ρ Density Dynamic iscosity Eddy-to-aminar iscosity ratio at the inet Mixture property ε Turbuent dissipation rate Fiter size in fiter-based mode Subscript i Interaction component Component Liquid L Laminar m Mixture property T Turbuent Vapor ω Free stream quantities Superscript Normaize aue tota Tota effect 9

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