A Generalized Compressible Cavitation Model

A new multi-phase model for low speed gas/liquid mixtures is presented; it does not require ad-hoc closure models for the variation of mixture density with pressure and yields thermodynamically correct acoustic propagation for multi-phase mixtures. The solution procedure has an interface-capturing scheme that incorporates an additional scalar transport equation for the gas void fraction. Cavitation is modeled via a finite rate source term that initiates phase change when liquid pressure drops below its saturation value. The numerical procedure has been implemented within a multi-element unstructured framework CRUNCH that permits the grid to be locally refined in the interface region. The solution technique incorporates a parallel, domain decomposition strategy for efficient 3D computations. Detailed results are presented for sheet cavitation over a cylindrical headform and a NACA 66 hydrofoil

Simulations of a Liquid Hydrogen Inducer at Low-Flow Off-Design Flow Conditions

The ability to accurately model details of inlet back flow for inducers operating a t low-flow, off-design conditions is evaluated. A sub-scale version of a three-bladed liquid hydrogen inducer tested in water with detailed velocity and pressure measurements is used as a numerical test bed. Under low-flow, off-design conditions the length of the separation zone as well as the swirl velocity magnitude was under predicted with a standard k-E model. When the turbulent viscosity coefficient was reduced good comparison was obtained a t all the flow conditions examined with both the magnitude and shape of the profile matching well with the experimental data taken half a diameter upstream of the leading edge. The velocity profiles and incidence angles a t the leading edge itself were less sensitive to the back flow length predictions indicating that single-phase performance predictions may be well predicted even if the details of flow separation modeled are incorrect. However, for cavitating flow situations the prediction of the correct swirl in the back flow and the pressure depression in the core becomes critical since it leads to vapor formation. The simulations have been performed using the CRUNCH CFD(Registered Trademark) code that has a generalized multi-element unstructured framework and a n advanced multi-phase formulation for cryogenic fluids. The framework has been validated rigorously for predictions of temperature and pressure depression in cryogenic fluid cavities and has also been shown to predict the cavitation breakdown point for inducers a t design conditions

Simulation of Cavitation Instabilities in Inducers

The cavitating performance of a sub-scale configuration of the SSME low pressure fuel pump (LPFP) has been simulated at off-design flow conditions where a back-flow vortex is generated at the leading edge. The numerical simulations have been compared with measured experimental data both for velocity profiles upstream of the inducer as well dynamic pressure traces on the shroud at the leading edge. Velocity profiles in the back-flow vortex for flow rates down to 70 percent of design were quantified; the swirl velocity comparisons were good while the axial velocity profile were reasonable but slightly over predicted the core velocity. Dynamic cavitating performance was modeled at a moderate Nss number of 20000 for 90 percent of design flow coefficient where rotational cavitation modes are present The source of this instability resulted from the interaction of the cavity with the neighboring blade leading to the detachment of the cavity that rotates relative to the blade and generates an asymmetric cavity pattern. The asymmetrical cavities generate a large radial load on the shaft which rotates at the fundamental mode of the rotational cavitation. For the sub-scale configuration the radial force amplitude was 186 lb-f which gives a non-dimensional force factor of 0.0116. Spectral analyses of the dynamic pressure traces on the shroud, at the leading edge plane, were compared with experimental measurements. The fundamental rotational cavitation mode was observed to be 125 Hz which is approximately 1.29 N (rotational frequency is 96. Hz); both the frequency and relative amplitude compared well with the unsteady measurements. In addition to the fundamental rotation cavitation mode the data shows substantial energy with multiple peaks in the 5 -7.5 N range. This range was reasonably represented in the numerical results although the spectrum was not as rich. A helical pressure wave at the fundamental mode is found to propagate upstream and a potential for interaction with structural elements was identified.http://deepblue.lib.umich.edu/bitstream/2027.42/84299/1/CAV2009-final122.pd

Efficient Evaluation of Low Degree Multivariate Polynomials in Ring-LWE Homomorphic Encryption Schemes

Homomorphic encryption schemes allow to perform computations over encrypted data. In schemes based on RLWE assumption the plaintext data is a ring polynomial. In many use cases of homomorphic encryption only the degree-0 coefficient of this polynomial is used to encrypt data. In this context any computation on encrypted data can be performed. It is trickier to perform generic computations when more than one coefficient per ciphertext is used. In this paper we introduce a method to efficiently evaluate low-degree multivariate polynomials over encrypted data. The main idea is to encode several messages in the coefficients of a plaintext space polynomial. Using ring homomorphism operations and multiplications between ciphertexts, we compute multivariate monomials up to a given degree. Afterwards, using ciphertext additions we evaluate the input multivariate polynomial. We perform extensive experimentations of the proposed evaluation method. As example, evaluating an arbitrary multivariate degree-3 polynomial with 100 variables over Boolean space takes under 13 seconds