A NEW MODELING OF SHEET CAVITATION CONSIDERING THE THERMODYNAMIC EFFECTS. Yumiko Sekino Ebara Cooporation, Ltd., Ota-ku, Tokyo JAPAN

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1 Cav3-GS-6-3 Fifth International Symposium on Cavitation (CAV3) Osaka, Japan, November -4, 3 A NEW MODELING OF SHEET CAVITATION CONSIDERING THE THERMODYNAMIC EFFECTS Takashi Tokumasu Institute of Fluid Science, Tohoku University Sendai, Miyagi JAPAN Yumiko Sekino Ebara Cooporation, Ltd., Ota-ku, Tokyo JAPAN Kenjiro Kamijo Institute of Fluid Science, Tohoku University Sendai, Miyagi JAPAN ABSTRACT In cryogenic fluids such as LH or LOX, temperature depression of liquids due to latent heat of vaporization suppresses the growth of cavity, which is called thermodynamic effects of cavitation. Thermodynamic effects of cavitation are significant in these fluids, because they are generally operated close to the critical point and are also characterized by a strong dependence of vapor pressure on temperature. Owing to this phenomenon, the performance of the rocket pump, inducer and other hydraulic equipment is not worse than would be predicted. In this paper, the thermodynamic effects of cavitation are investigated numerically. To predict these effects, a cavitation model introduced by Deshpande et al. is improved. Using this model a sheet cavity around a -D hydrofoil is simulated and the dependence of the thermodynamic properties of fluids or Reynolds number on the thermodynamic effects of cavitation are analyzed. The numerical results explain the thermodynamic effects well. INTRODUCTION These days, satellite systems such as satellite broadcasting, and navigation by GPS are becoming indispensable for our life. Liquid rockets are mainly used to launch the satellites. In the rockets, liquid hydrogen (LH )and liquid oxygen (LOX) are used as propellants. A turbopump which supplies the propellants to a combustion chamber with high pressure is incorporated in the rocket engine to make the system smaller and lighter. An inducer is attached to the turbopump to increase the efficiency of the pump. It is necessary for the turbopump to run very fast to make it smaller.[] In this condition, cavitation occurs around the inducer because the local static pressure becomes smaller than the vapor pressure and it makess the efficiency of the turbopump worse. In general, the pressure observed inside the cavity of cryogenic fluids is lower than the vapor pressure corresponding to the free stream temperature. The reason for this is that the temperature of the liquid in the vicinity of the cavity surface is depressed below the free stream temperature because the latent heat of vaporization must be extracted from the bulk liquid. Due to the strong dependence of the vapor pressure on the temperature, the cavity size in cryogenic fluids is smaller than would be predicted if there were no temperature depression. This phenomenon is called the thermodynamic effects of cavitation. This phenomenon sustains the efficiency of the turbopump. Analyses of thermodynamic effects have been conducted focusing mainly on the degree of temperature depression as a function of flow conditions and liquid properties. Stepanoff defined a parameter which expressed the effect of thermodynamic properties of liquid on cavity volume.[] Ruggeri expressed the correlation between the liquid which contributes to the evaporation and the cavity volume by an equation obtained experimentally.[3] Kato introduced the Z factor theory about the pressure depression due to the thermodynamic effects.[4] However, in spite of previous analyses, the detailed mechanism of the thermodynamic effects of cavitation has not been clearly understood. Recently, Deshpande et al. presented a numerical modeling of the thermodynamic effects of cavitation and applied it to the flow around a hydrofoil.[5] However, the cavitation model is not considered to be able to express the real phenomenon accurately. In this study, the cavitation model introduced by Deshpande et al. is improved and is applied to the flowfield around a hydrofoil to analyze the thermodynamic effects of cavitation. The analysis is restricted to partial sheet cavitation. A Navier Stokes solver based on artificial compressibility and pseudo time stepping coupled with an energy equation is used to simulate the thermal boundary layer in the fluid from which details of the thermodynamic effects and temperature depression can be obtained. The cavity surface is defined as the streamline from the inception point of the sheet cavity. This model is applied to the cavitating flow of water and liquid oxygen, and the thermodynamic effects of cavitation of liquid oxygen is estimated. Moreover, the effect of Reynolds number on the thermodynamic effects of cavitation is analyzed. NOMENCLATURE

2 C p pressure coefficient L latent heat of vaporization p pressure Pr Prandtl number Re Reynolds number t time T temperature U upstream flow velocity u, v flow velocity in x, y direction σ cavitation number ρ density λ thermal conductivity τ curvature κ surface tension coefficient µ viscosity coefficient NUMERICAL METHOD In this paper, the complete flow and temperature fields are described by the incompressible, two dimensional Navier Stokes equations and energy equation as follows.[5] Continuity Equation: u x + v y = () Two dimensional Navier Stokes Equations: u t + u x + uv y + p x = Re v t + uv x + v y + p y = Re ( u x + u y ( v x + v y Energy equation: T t + u T x + v T y = { ( ) T Re Pr x + T y ( ) u ( ) v ( u x y y + v ) } (4) x In this paper, the artificial compressibility term is added to Eq.() to enable a time marching scheme to advance the flow variables in the system. The above equation becomes p β t + u x + v =. (5) y When the solution converges, the time derivatives in the equations approach zero and the steady state Navier Stokes solution is obtained. In this paper, the value of β is considered to be.5. Equations () (3), (4) and (5) are changed to generalized coordinates and solved by the Finite Differential Method. The advection term is estimated by LAX and the nd order MUSCL scheme and the viscous term is estimated by nd order central differencing. Local time stepping is used to improve convergence and the time derivative is solved by the LU-SGS method. The solver is applied to the flow over a NACA hydrofoil. The geometry of the hydrofoil is shown in Fig.. ) ) () (3) Figure : C-grid used in Navier Stokes computations for cavitating flow over NACA hydrofoil. Grid size is 35 5 The number of the grid is for the upper part of the hydrofoil, for the lower part of the hydrofoil, for the wake and 5 for the part normal to the hydrofoil in order to express a cavity adjacent to the upper body surface. The thermal boundary layer above the hydrofoil and a cavity must be captured to calculate the thermodynamic effects of cavitation accurately. The minimum grid size isassumedtobe 4 of the chord length by considering that the flow Reynolds number is from 6 to 7. The simulation region is five times as large as the chord length. The boundary conditions of the simulation are set as listed in Table. The boundary conditions of the cavity are given in the next subsection. In this simulation, the Table : Boundary conditions pressure velocity temperature solid Neumann fixation wall condition wake average of upper and lower value inflow Neumann fixation boundary condition outflow fixation order extrapolation boundary Reynolds number, Re, is determined based on the chord length, D, as Re = ρ lu D, (6) µ m where µ m is the molecular viscosity. However, the turbulent viscosity, µ t, must be considered because of the high Reynolds number. In this simulation, the Reynolds

3 number, Re, in Eqs.() (4) is replaced by Re = ρ lu D µ m + µ t (7) length of the afterbody is. times that of the cavity.[7] Details of the typical deformed grid around the cavity profile is shown in Fig.. and this turbulent viscosity is estimated by applying the algebraic Baldwin Lomax turbulence model.[6] CAVITATION MODEL In this paper, a cavity surface is defined by the streamline from the inception point of the sheet cavity. However, since neither the cavity surface nor its inception point is known prior to the solution, an iterative procedure is needed to determine them. First, the point of the hydrofoil is switched from a solid wall point to a cavity point if the pressure drops below the local vapor pressure. On the cavity surface, liquid pressure is obtained by the Neumann condition and gas pressure is obtained as the vapor pressure corresponding to the local temperature. The tangential velocity of the interface is determined by the Neumann condition. Normal velocity of the interface is allowed to deviate from zero in order to express the growth of the cavity. In the paper of Deshpande et al., the normal velocity is determined by the Method of Characteristics, but this condition cannot express the pressure balance on the cavity surface.[5] For this reason the normal velocity on the cavity surface is given by G = k (p g p l κτ), (8) where k is the proportional constant, κ is the surface tension coefficient and τ is the curvature of the interface. In this paper, k isassumedtobe. The growth of the cavity stops if the pressure differences between gas and liquid equals the surface tension. When the flow reaches steady state, the pressure balance between the gas pressure, p g, liquid pressure p l and surface tension, κτ, is achieved. A new cavity surface at the next step is obtained by tracing the streamline from the inception point of the cavity to the last cavitating grid. The inception point is determined as the first grid on the hydrofoil at which the pressure is less than the local vapor pressure. The computational domain is re gridded accordingly over the hydrofoil/cavity surface.[5] In this paper, grid updating is accomplished every steps. If the streamline enters the hydrofoil, the location of the interface in the next time step is determined on the hydrofoil. In general, the cavity ends with a finite thickness, and therefore the cavity must be closed by adding an afterbody. The afterbody shape used in this paper is a cubic profile that merges smoothly with the cavity interface and hydrofoil. On the afterbody, solid boundary conditions are used except for the tangential velocity. The tangential velocity is obtained by U = U s x x s x f x s, (9) where x s is the x coordinate of the end of the cavity, x f is that of the end of the afterbody, and U s is the tangential velocity at the end of the cavity.[7] In this paper, the Figure : Details of a typical deformed grid The boundary conditions of temperature on the cavity surface are obtained from an energy balance at the interface mentioned below.[5] As shown in Fig.3, the vapor is considered to evaporate by the volume flow rate of dq from the elemental surface area on the interface, ds. The heat flux required to vaporize the fluid from an elemental portion of the cavity surface is given by q l ds = ρ v LdQ, () where q l is the heat flux, ρ v is the density of the vapor and L is the latent heat of the liquid. The heat conducted to the cavity surface is given by dt q l = λ l dn, () where n is the normal vector to the liquid vapor interface. From Eqs.() and (), the temperature gradient at the liquid vapor interface is given by dt dn = ρ vl λ l dq ds. () The computation of the energy balance in Eq.(), however, requires knowledge of the rate of vaporization, dq, from the elemental surface area on the interface. However, it is difficult to model dq accurately without the knowledge of the flow field inside the cavity. In this paper, therefore, the rate of vaporization, dq, is modeled from the mass balance inside the cavity. If the velocity gradient inside the cavity is assumed to be constant as shown in Fig.3, one obtains dq = u n+h n+ u nh n = d(uh), (3) where h is the length between the cavity surface and the hydrofoil. Equation (3) is inserted into Eq.() and Eq.() is reduced. Moreover, du for simplicity and stability of the simulation, one obtains dt dn = RePr ρ v Lu dh ρ l U ds. (4) This equation is used as the boundary condition of temperature on the cavity surface. In Eq.(4), Prandtl number, Pr, density of gas and liquid, ρ v and ρ l, respectively, and latent heat, L, arethe 3

4 Liquid Cavity u n ds dq h n+ h n u n+ Hydrofoil Figure 3: Temperature boundary Cavity Profile : LOX (8 K) : LOX (9 K) : Water (9 K) Table : Parameters of the flowfield Prandtl Number Pr Viscosity Coef. µ [µpa s] Latent Heat L [kj/kg] Isobaric Specific Heat C p [kj/kgk] Surface Tension Coef. σ [mn/m] LOX LOX Water 8 K 9 K 9 K Temperature Depression [K] Figure 4: Cavity profile of each liquid : LOX (8 K) : LOX (9 K) : Water (9 K) thermodynamic properties and depend on the temperature. In this paper, these thermodynamic properties except for the density of the gas and liquid is assumed to be constant because they are changed smaller compared with the density. Therefore, Eq.(4) must be solved by an iterative process. While a large portion of the vapor breaks off from the cavity in reality, all the vapor is assumed to be condensed in this study. Therefore, the temperature at the end of the cavity is overestimated compared with the real case. For this reason, the temperature rise due to condensation is not considered in this paper. RESULTS AND DISCUSSTIONS First, this computer program is applied to the flowfield of water at T = 9 K, and to that of liquid oxygen at T = 8 K and T = 9 K. The Reynolds number is 7. The parameters of the flowfield are shown in Table. The attack angle of the hydrofoil is four degrees. Figures 4, 5 and 6 show the shape of the cavity, the temperature distribution and the pressure coefficient of each fluid, respectively. As shown in Fig.4, the cavity in water at T = 9 K is larger than that in liquid oxygen. Even when the vapor pressure is fixed to that at the free stream temperature in the case of water, the size of the cavity is almost the same as that shown in Fig.4. Therefore, it can be said that the thermodynamic effects of cavitation in water are small and that the size of the cavity does not Figure 5: Temperature depression of each liquid change greatly due to the thermodynamic effects of cavitation. On the other hand, the thermodynamic effects of cavitation in liquid oxygen are greater than those in water, and therefore it is observed that the size of the cavity decreases greatly. Two causes of the difference of the thermodynamic effects of cavitation in these liquids are considered. First, while the ratio of gas density to liquid density of water is /7 at T = 9 K, that of liquid oxygen is /8 at T = 8 K. The temperature depression, therefore, is greater in liquid oxygen than in water as shown by Eq.(4), which is observed in Fig.5. Moreover, even if the temperature depressions in these liquids are the same, the vapor pressure depression in liquid oxygen is larger than that in water because liquid oxygen is operated close to the critical temperature. Owing to these effects, the size of the cavity in liquid oxygen is smaller than that in water. Moreover, due to the above mentioned reason, it is considered that the thermodynamic effects of cavitation in liquid oxygen at T = 9 K are greater than those at T = 8 K; this phenomenon is observed in Fig.4. The temperature depression on the cavity surface, however, is smaller at T = 9 K than at T = 8 K, because the size of the cavity at T = 9 K is smaller, and therefore the value dh/ds in Eq.(4) is smaller. The increase of the 4

5 Pressure Coefficient, -Cp Non-cavitating(LOX, 9K) : LOX (8 K) : LOX (9 K) : Water (9 K) Cavity Profile : Re=.E6 : Re=5.E6 Figure 6: Pressure coefficient of each liquid Figure 7: Cavity profile (LOX) thermodynamic effects of cavitation with the increase of liquid temperature is observed experimentally and qualitatively agrees with the simulation results. From Fig.6 it is observed that the value, C p,onthe cavity region in liquid oxygen at T = 8 K is larger than that in water, even though their cavitation numbers are the same. The pressure coefficient, C p,isdefinedas C p = p l p. (5) ρ lu The minimum cavitation number, σ min, in which the vapor pressure depression due to the evaporation is considered is expressed as σ min = p p vmin ρ lu. (6) In this study the relation p l = p v is established when the steady state is achieved by neglecting the effect of surface tension. In this condition, C p = σ min is established at the point where the pressure on the cavity surface is lowest from Eqs.(5) and (6). The vapor pressure on the cavity surface, p v, decreases due to the evaporation, σ min increases, and therefore C p which equals σ min increases. Moreover, C p becomes larger in liquid oxygen at T = 9 K than that of noncavitating flow. This is because the velocity on the cavity surface is greater than that on the hydrofoil in noncavitating flow, and therefore the static pressure decreases. Next, the effect of the Reynolds number on the cavity size is analyzed. The temperature of liquid oxygen is 8 K and the cavitation number is.9. Other parameters are shown in Table. The Reynolds numbers of. 7 and.5 7 are chosen. First, the shapes of the cavity in each flowfield are shown in Fig.7 and the temperature depression in each flowfield without the thermodynamic effects of cavitation are shown in Fig.8, where the sizes of the cavities are the same due to the same cavitation number. As shown in Fig.7, the size of the cavity at a high Reynolds number is larger than that at a low Reynolds Temperature Depression [K] : Re=.E6 : Re=5.E Figure 8: Temperature distribution (without therm.effects) number, and therefore it can be said that the thermodynamic effects of cavitation decrease as the Reynolds number increases. As shown in Fig.8, however, the temperature depression at the high Reynolds number is larger than that at the low Reynolds number, even though the sizes of the cavity are the same. It may therefore be predicted that the thermodynamic effects of cavitation at the higher Reynolds number are expected to be larger than those at the lower Reynolds number, contrary to the results of Fig.7. The trend of these figures can be explained by considering that the thermodynamic effects of cavitation depend not only on the vapor pressure depression due to the temperature depression but also on the velocity of the free stream. The increase of the Reynolds number corresponds to the increase of free stream velocity by U = Reµ ρ l D, (7) when the kinds of liquid are the same. Therefore, the free stream pressure, p, is larger at a higher Reynolds number even when the cavitation number of the flow is the same. Using the free stream cavitation number, σ,the 5

6 minimum cavitation number, σ min, is expressed by σ min = p p vmin ρ lu = σ + p v, (8) ρ lu where p v = p v p vmin is the pressure depression due to the evaporation. The simulation results at Re =. 7 and Re =.5 7 are shown in Table 3. The Table 3: The cavitation number σ min against the Reynolds number. Reynolds number Re temperature T [K] 8 8 density at T = 8 K ρ l [kg/m 3 ] inflow pressure p [MPa].96.9 velocity U [m/s].. saturated pressure p v [MPa].3.3 temperature high low local T vmin [K] (75) (57) pressure high low p vmin [MPa] pressure depression p v [MPa] p v ρ lu cavitation number σ min (.45) small (.56) large (.) large (.) (.3) large (.98) small (.) small (.99) value T vmin is the minimum temperature on the cavity surface. From Table 3, the pressure depression due to the evaporation, p v, is larger at the high Reynolds number (Re =. 7 ) than that at the low Reynolds number (Re =.5 7 ). Because of the difference of U, however, the second term of Eq.(8), p v /ρ l U,is smaller at the high Reynolds number, and therefore the minimum cavitation number, σ min, is smaller. Namely, at a high Reynolds number, the pressure depression due to the evaporation compared to the dynamic pressure, p v /ρ l U, is relatively small. For this reason the minimum cavitation number, σ min, is smaller at a higher Reynolds number and the size of the cavity is larger. and the normal velocity on the cavity surface was allowed to deviate from to simulate the growth of the cavity. The boundary condition of energy equation on the cavity interface was determined by the energy balance of the evaporation process. This model was applied to the two dimensional flow around a hydrofoil and the size of the cavity was compared by changing the kind of liquid and the Reynolds number. Consequently, it was qualitatively estimated that the thermodynamic effects of cavitation in water is smaller than that of liquid oxygen because the ratio between liquid and vapor is larger in water and because of the small dependence of the temperature on the vapor pressure. Moreover, it was found that the thermodynamic effects of cavitation at a higher Reynolds number were suppressed because the pressure depression due to the evaporation is relatively small compared to the dynamic pressure. References [] Kamijo, K., Yoshida, M. and Tsujimoto, Y., Hydraulic and Mechanical Performance of LE-7 LOX Pump Inducer, AIAA Journal of Propulsion and Power,Vol.9, (993), pp [] Stahl, H.A. and Stepanoff, A.J., Thermodynamic Aspects of Cavitation in Centrifugal Pumps, Trans. ASME, 78 8, (956) [3] Ruggeri, R.S. and Moore, R.D., Method for Prediction of Pump Cavitation Performance for Various Liquids, Liquid Temperatures, and Rotative Speeds, NASA TN D-59, (969) [4] Kato, H., Thermodynamic Effect on Incipient and Developed Sheet Cavitation, International Symposium on Cavitation, ASME, (984) [5] Deshpande, M., Feng, J. and Merkle, C.L., Numerical Modeling of Thermodynamic Effects of Cavitation, ASME Journal of Fluids Engineering, Vol. 9, (997), pp [6] Baldwin, B. and Lomax, H., Thin-Layer Approximation and Algebraic Model for Separated Turbulent Flows, AIAA Paper 78-57, (978) [7] Kinnas, S.A. and Fine N.E., A Numerical Nonlinear Analysis of the Flow Around Two- and Threedimensional Partially Cavitating Hydrofoils J. Fluid Mech. 54 (993), pp.5 8 CONCLUDING REMARKS In this paper, a method introduced by Deshpande et al. to estimate the size of sheet cavitation with the thermodynamic effects of cavitation was improved and the thermodynamic effects of cavitation were analyzed. A Navier Stokes solver based on artificial compressibility and pseudo time stepping coupled with the energy equation was used to simulate the thermodynamic effects of cavitation. Cavity surface at the next time step was determined by tracing the streamline from the inception point, 6

DEVELOPMENT AND VALIDATION OF NEW CRYOGENIC CAVITATION MODEL FOR ROCKET TURBOPUMP INDUCER

Proceedings of the 7 th International Symposium on Cavitation CAV9 Paper No. 63 August 7-, 9, Ann Arbor, Michigan, USA DEVELOPMENT AND VALIDATION OF NEW CRYOGENIC CAVITATION MODEL FOR ROCKET TURBOPUMP