A dual-time implicit preconditioned Navier-Stokes method for solving 2D steady/unsteady laminar cavitating/noncavitating flows using a Barotropic model

A two-dimensional, time-accurate, homogeneous multiphase, preconditioned Navier-Stokes method is applied to solve steady and unsteady cavitating laminar flows over 2D hydrofoils. A cell-centered finite-volume scheme employing the suitable dissipation terms to account for density jumps across the cavity interface is shown to yield an effective method for solving the multiphase Navier-Stokes equations. This numerical resolution is coupled to a single-fluid model of cavitation that the evolution of the density is governed by a barotropic sate law. A preconditioning strategy is used to prevent the system of equations to be stiff. A dual-time implicit procedure is applied for time accurate computation of unsteady cavitating flows. A sensitivity study is conducted to evaluate the effects of various parameters such as numerical dissipation coefficients and preconditioning on the accuracy and performance of the solution. The computations are presented for steady and unsteady laminar cavitating flows around the NACA0012 hydrofoil for different conditions. The solution procedure presented is shown to be accurate and efficient for predicting steady/unsteady laminar cavitating/noncavitating flows over 2D hydrofoils.http://deepblue.lib.umich.edu/bitstream/2027.42/84312/1/CAV2009-final138.pd

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

A robust inverse design solver for controlling the potential aggressiveness of cavitating flow on hydrofoil cascades

This article presents the development of a new inverse design algorithm capable of generating blade geometries for cavitating cascade flows. With this methodology, we demonstrate the controllability of the pressure distribution in and around the cavity and thereby provide a means to regulate the aggressiveness of blade cavitation phenomena. The solver proposed here uses the Tohoku–Ebara equation of state to model phase change, combined with bespoke preconditioning and multigrid methods designed to handle the system's ill conditioning and cope with the hypersonic flow regime of the mixture, respectively. Blade geometries and the cavitating flow field are calculated simultaneously in a robust and efficient manner, with a blade loading that matches the target distribution. In this article, the accuracy of the cavitating flow solver is first demonstrated for the NACA0015 hydrofoil case and associated experimental data. The inverse design procedure is then applied to a typical axial flow pump cascade: a new blade profile is generated with a topology that successfully reduces the gradient of the pressure jump at cavity closure

Novel Blade Design Strategy to Control the Erosion Aggressiveness of Cavitation

With the reduction in size of turbomachinery systems, cavitation aggressiveness is intensified. Erosion, caused by the repeated collapse of gaseous bubbles in proximity to solid surfaces, occurs at rates that dramatically downgrade the life expectancy of rotating parts. As a result, the compacting strategy, meant to reduce cost and improve efficiency, fails for liquid flows. The research undertaken here proposes a novel design method aimed at controlling the erosion aggressiveness of cavitation. The underlying idea is that the cavity closure shock is a determining factor in the intensity of bubble collapse mechanisms: sharp and high amplitude shocks give rise to strong erosion, while low gradient and low amplitude recoveries reduce the erosive intensity. The working hypothesis is tested here, first, by developing a novel inverse design algorithm capable of handling cavitating flow. The code solves the inviscid Euler equations and models blade cavitation using the Tohoku-Ebara barotropic equation of state. Bespoke preconditioning and multigrid procedures are constructed to handle the large amplitudes in flow regime (from hypersonic in the cavity to very low Mach number in the liquid phase). The inverse solver is then used to produce a set of 2D cascade hydrofoil geometries with smoothed shock profiles at cavity closure. The blades are assessed numerically using both steady state and time-resolved approaches. Both hydrodynamic performance, given in terms of swirl, lift and drag, and cavitation dynamics are evaluated. Recently developed erosion prediction methodologies are implemented and demonstrate compelling correlations between the erosion patterns and shock profile. Finally, experimental testing is carried out using a purposefully developed observation platform. The erosive performance of two of the geometries is measured using the paint removal technique. Results reveal a significant improvement in erosive response for the shock smoothed design, thus confirming the numerical findings as well as the validity of the design hypothesis

Implicit preconditioned numerical schemes for the simulation of three-dimensional barotropic flows

A numerical method for simulating three-dimensional, generic barotropic flows on unstructured grids is developed. Space and time discretizations are separately considered. A finite volume compressible approach, based on a suitable Roe numerical flux function, is proposed and the accuracy of the resulting semi-discrete formulation for nearly-incompressible flows is ensured by ad hoc preconditioning. Moreover, a linearized implicit time-advancing technique is proposed, only relying on the algebraic properties of the Roe flux function and therefore applicable to a variety of problems. This implicit strategy is extended so as to incorporate the aforementioned preconditioning. The considered numerical ingredients are firstly defined in a one-dimensional context; after validation, they are extended to three-dimensional non-rotating as well as rotating frames. Finally, the resulting numerical method is validated by considering complex industrial flows, namely the water flow around a hydrofoil (for which specific experimental data are available) and the water flow around a rotating turbo-pump inducer. By starting from a particular industrial problem (namely the numerical simulation of propellant flows around an axial inducer belonging to the feed turbo-pump system of a liquid propellant rocket engine), a numerical method which can be applied to generic barotropic flows is defined. Along the way, a constructive procedure for solving the 1D Riemann problem associated with a generic convex barotropic state law is proposed. This solution, also exploited for defining a Godunov numerical flux suitable for incorporation into finite volume schemes, is systematically used in order to define exact benchmarks for the quantitative validation of the proposed one-dimensional numerical methods

Investigation of three-dimensional effects on a cavitating Venturi flow

International audienceA numerical investigation of the behaviour of a cavitation pocket developing along a Venturi geometry has been performed using a compressible one-fluid hybrid RANS/LES solver. The interplay between turbulence and cavitation regarding the unsteadiness and structure of the flow is complex and not well understood. This constitutes a determinant point to accurately simulate the dynamic of sheet cavities. Various turbulent approaches are tested: a new Scale-Adaptive model and the Detached Eddy Simulation. 2D and 3D simulations are compared with the experimental data. An oblique mode of the sheet is put in evidence

Numerical Simulations of Barotropic Flows in Complex Geometries

Nella presente tesi é considerata la simulazione di flussi barotropici all'interno di geometrie complesse e le sue applicazioni a flussi cavitanti e a problemi di trasporto di sedimenti. Un approccio generale basato su vari metodi ai volumi finiti e applicabile a griglie non strutturate, é stato sviluppato e testato. L'estensione al secondo ordine spaziale é ottenuta usando metodi tipo MUSCL. L'avanzamento temporale si basa su metodi impliciti linearizzati e il secondo ordine temporale si basa sulla tecnica Defect Correction. Per quanto riguarda i flussi cavitanti, é stato utilizzato un modello a flusso omogeneo che si basa su una equazione di stato barotropica. Il bilancio di massa e di quantità di moto per flussi comprimibili sono discetizati tramite un metodo misto ai volumi finiti e agli elementi finiti. Gli elementi finiti P1 sono utilizzati per i termini viscosi mentre i volumi finiti per quelli convettivi. I flussi numerici sono calcolati utilizzando schemi in grado di calcolare soluzioni discontinue e una strategia di precondizionamento ad-hoc é stata utilizzata per risolvere i problemi di accuratezza che si riscontrano per bassi numeri di Mach. Una funzione di flusso di tipo HLL per flussi barotropici é stata proposta introdotta. In questa funzione di flusso é stato aggiunto un termine antidiffusivo che riduce i problemi di accuratezza che tipicamente si riscontrano per discontinuità di contatto e flussi viscosi quando si utilizzano schemi appartenenti a questa categoria. Per questa funzione di flusso di classe HLL due differenti linearizzazione temporali sono state considerate: nella prima la matrice di upwind della funzione di flusso é considerata constante, mentre nella seconda la sua variazione temporale viene tenuta in considerazione. Gli ingredienti numerici proposti sono stati quindi testati simulando varie tipologie di flussi, in particolare lo strato limite di Blasius, un problema di Riemann, il flusso quasi-1D in un ugello e il flusso di acqua intorno ad un profilo, sia in condizioni cavitanti che non cavitanti. Inoltre l'introduzione degli effetti della turbolenza tramite il modello RANS k-epsilon é stata testata simulando il flusso ad alto numero di Reynolds su una lastra piana e, per finire, é stata affrontata la simulazione numerica di un induttore reale tridimensionale, sia in condizioni non cavitanti e cavitanti. Oltre a quanto detto sono stati considerati anche problemi di trasporto di sedimenti. Il modello fisico di questo problema é basato sulle equazioni Shallow-Water a cui si aggiunge l'equazione di Exner per descrivere l'evoluzione temporale del profilo del fondale. In particolare é il flusso di sedimenti é stato descritto utilizzando il modello di Grass. Il sistema completo di equazioni é stato discretizzato utilizzando due metodi ai volumi finiti, lo schema di tipo predittore-correttore SRNH e uno schema di Roe modificato per sistemi di equazioni in forma non conservativa. Partendo dalle versioni esplicite di questi schemi, sono stati sviluppati i corrispondenti metodi impliciti e, in particolare lo Jacobiano della funzione di flusso é stato calcolato utilizzando strumenti di differenziazione automatica. Questo approccio permette di non dover calcolare manualmente le complesse espressioni delle derivate della funzione di flusso. Questi metodi sono poi stati comparati in termini di accuratezza e costi computazionali utilizzando specifici problemi monodimensionali e bidimensionali caratterizzati da scale temporali diverse per l'evoluzione del fondale e del flusso d'acqu

Preconditioning methods for ideal and multiphase fluid flows

The objective of this study is to develop a preconditioning method for ideal and multiphase multispecies compressible fluid flow solver using homogeneous equilibrium mixture model. The mathematical model for fluid flow going through phase change uses density and temperature in the formulation, where the density represents the multiphase mixture density. The change of phase of the fluid is then explicitly determined using the equation of state of the fluid, which only requires temperature and mixture density. The method developed is based on a finite-volume framework in which the numerical fluxes are computed using Roe’s [1] approximate Riemann solver and the modified Harten, Lax and Van-leer scheme (HLLC) [2]. All speed Roe and HLLC flux based schemes have been developed either by using preconditioning or by directly modifying dissipation to reduce the effect of acoustic speed in its numerical dissipation when Mach number decreases. Preconditioning proposed by Briley, Taylor and Whitfield [3], Eriksson [4] and Turkel [5] are studied in this research, where as low dissipation schemes proposed by Rieper [6] and Thornber, Mosedale, Drikakis, Youngs and Williams [7] are also considered. Various preconditioners are evaluated in terms of development, performance, accuracy and limitations in simulations at various Mach numbers. A generalized preconditioner is derived which possesses well conditioned eigensystem for multiphase multispecies flow simulations. Validation and verification of the solution procedure are carried out on several small model problems with comparison to experimental, theoretical, and other numerical results. Preconditioning methods are evaluated using three basic geometries; 1) bump in a channel 2) flow over a NACA0012 airfoil and 3) flow over a cylinder, which are then compared with theoretical and numerical results. Multiphase capabilities of the solver are evaluated in cryogenic and non-cryogenic conditions. For cryogenic conditions the solver is evaluated by predicting cavitation on two basic geometries for which experimental data are available, that is, flow over simple foil and a quarter caliber hydrofoil in a tunnel using liquid nitrogen as a fluid. For non-cryogenic conditions, water near boiling conditions is used to predict cavitation on two simple geometries, that is, flow over simple foil in a tunnel and flow over a one caliber ogive. Cavitation predictions in both cryogenic and non-cryogenic cases are shows to agree well with available experimental data

Adjoint-based geometry optimisation with applications to automotive fuel injector nozzles

Methods of Computational Fluid Dynamics (CFD) have matured, over the last 30 years, to a stage where it is possible to gain substantial insight into fluid flow processes of engineering relevance. However, the motives of fluid dynamic engineers typically go well beyond the level of improved understanding, to the pragmatic aim of improving the performance of the engineering systems in consideration. It is in recognition of these circumstances that the present thesis investigates the use of automated design optimisation methodologies in order to extend the power of CFD as an engineering design tool. Optimum design problems require the merit or performance of designs to be measured explicitly in terms of an objective function. At the same time, it may be required that one or more constraints should be satisfied. To describe allowable variations in design, shape parameterisation using basic geometric entities such as straight lines and arcs is employed. Taking advantage of previous experience in the research group concerning cavitating flows, a fully automated method for nozzle design/optimisation was developed. The optimisation is performed by means of discharge coefficient (Cd) maximisation. The objective is to design nozzle hole shapes that maximise the nozzle Cd for a given basic nozzle geometry (i.e. needle and sac profile) and reduce or even eliminate the negative pressure region formed at the entry of the injection hole. The deterministic optimisation model was developed and implemented in the in-house RANS CFD code to provide nozzle shapes with pre-defined flow/performance characteristics. The required gradients are calculated using the continuous adjoint technique. A parameterisation scheme, suitable for nozzle design, was developed. The localised region around the hole inlet, where cavitation inception appears, is parameterised and modified during the optimisation procedure, while the rest of the nozzle remains unaffected. The parameters modifying the geometry are the radius of curvature and the diameter of the hole inlet or exit as well as the relative needle seat angle. The steepest descent method has been used to drive the calculated gradients to zero and update the design parameters. For the validation of the model two representative inverse design cases have been selected. Studies showing the behaviour of the model according to different numerical and optimisation parameters are also presented. For the purpose of optimising the geometries, a cost function intended to maximise the discharge coefficient was defined. At the same time it serves the purpose of restructuring geometries which have controlled or eliminated cavitation inception in the hole entrance. This is identified in the steady-state mode by reduction of the volume of negative relative pressure appearing in the hole entrance. Results of cavitation control on some representative nozzle geometries show significant benefits gained by the use of the developed method. This is mainly because the developed model performs optimisation on numerous parametric combinations automatically. Results showed that, by using the proposed method, geometries with larger Cd values can be achieved and the cavitation inception can, in some cases, be completely eliminated. Cases where all the parameters were combined for redesign the geometry required less modification to predict larger Cd values than cases where each parameter was modified individually. This is an important result since manufacturers are seeking improvement in the performance of products resulting from the least geometry modifications