Modeling of turbulent,isothermal and cryogenic cavitation under attached conditions

Cavitation is often triggered when the fluid pres- sure is lower than the vapor pressure at a local thermo- dynamic state. The present article reviews recent progress made toward developing modeling and computational strat- egies for cavitation predictions under both isothermal and cryogenic conditions, with an emphasis on the attached cav- ity. The review considers alternative cavitation models along Reynolds-averaged Navier-Stokes and very lager eddy simu- lation turbulence approaches to ensure that the computational tools can handle flows of engineering interests. Observing the substantial uncertainties associated with both modeling and experimental information, surrogate modeling strategies are reviewed to assess the implications and relative impor- tance of the various modeling and materials parameters. The exchange between static and dynamic pressures under the influence of the viscous effects can have a noticeable impact on the effective shape of a solid object, which can impact the cavitation structure. The thermal effect with respect to evaporation and condensation dynamics is examined to shed light on the fluid physics associated with cryogenic cav- itation. The surrogate modeling techniques are highlighted in the context of modeling sensitivity assessment. Keywords

NUMERICAL SIMULATION OF 3-D TURBULENT FLOWS OVER DREDGED TRENCHES

A 3- D free surface flow in open channels based on the Reynolds equations with the k-ε turbulence closure model is presented in this paper. Insted of the 'rigid lid' approximation, the solution of the free surface equation is implemented in the velocity-pressure iterative procedure on the basis of the conventional SIMPLE method. This model was used to compute the flow in rectangular channels with trenches dredged across the bottom. The velocity, eddy viscosity coefficient, turbulent shear stress, turbulent kinetic energy and elevation of the free surface can be obtained. The computed results are in good agreement with previous experimental data.

Multi-scale analysis of subgrid stress and energy dissipation in turbulent channel flow

In present study, the subgrid scale （SGS） stress and dissipation for multiscale formulation of large eddy simulation are analyzed using the data of turbulent channel flow at Ret = 180 obtained by direct numerical simulation. It is found that the small scale SGS stress is much smaller than the large scale SGS stress for all the stress components. The dominant contributor to large scale SGS stress is the cross stress between small scale and subgrid scale motions, while the cross stress between large scale and subgrid scale motions make major contributions to small scale SGS stress. The energy transfer from resolved large scales to subgrid scales is mainly caused by SGS Reynolds stress, while that between resolved small scales and subgrid scales are mainly due to the cross stress. The multiscale formulation of SGS models are evaluated a priori, and it is found that the small- small model is superior to other variants in terms of SGS dissipation.

APPLICATION OF A MODIFIED QUICK SCHEME TO DEPTH-AVERAGED k-ε TURBULENCE MODEL BASED ON UNSTRUCTURED GRIDS

The modified QUICK scheme on unstructured grid was used to improve the advection flux approximation, and the depth-averaged κ-ε turbulence model with the scheme based on FVM by SIMPLE series algorithm was established and applied to spur-dike flow computation. In this model, the over-relaxed approach was adopted to estimate the diffusion flux in view of its advantages in reducing errors and sustaining numerical stability usually encountered in non-orthogonal meshes. Two spur-dike cases with different defection angles （90°and 135°） were analyzed to validate the model. Computed results show that the predicted velocities and recirculation lengths are in good agreement with the observed data. Moreover, the computations on structured and unstructured grids were compared in terms of the approximately equivalent grid numbers. It can be concluded that the precision with unstructured grids is higher than that with structured grids in spite that the CPU time required is slightly more with unstructured grids Thus, it is significant to apply the method to numerical simulation of practical hydraulic engineering.

THREE-DIMENSIONAL SIMULATION OF MEANDERING RIVER BASED ON 3-D RNG k-ε TURBULENCE MODEL

A 3-D numerical model for calculating flow in non-curvilinear coordinates was established in this article. The flow was simulated by solving the full Reynolds-averaged Navier-Stokes equations with the RNG κ-ε turbulence model. In the horizontal x-y-plane, a boundary-fitted curvilinear co-ordinate system was adopted, while in the vertical direction, a σ co-ordinate transformation was used to represent the free surface and bed topography. The water level was determined by solving the 2-D Poisson equation derived from 2-D depth averaged momentum equations. The finite-volume method was used to discretize the equations and the SIMPLEC algorithm was applied to acquire the coupling of velocity and pressure. This model was applied to simulate the meandering channels and natural rivers, and the water levels and the velocities for all sections were given. By contrasting and analyzing, the agreement with measurements is generally good. The feasibility studies of simulating flow of the natural fiver have been conducted to demonstrate its applicability to hydraulic engineering research.

A New Efficient Finite Volume Modeling of Small Amplitude Free Surface Flows with Unstructured Grid

A staggered finite-volume technique for non-hydrostatic, small amplitude free surface flow governed by the incompressible Navier-Stokes equations is presented there is a proper balance between accuracy and computing time. The advection and horizontal diffusion terms in the momentum equation are discretized by an integral interpolation method on the orthogonal unstructured staggered mesh and, while it has the attractive property of being conservative. The pressure-correction algorithm is employed for the non-hydrostatic pressure in order to achieve second-order temporal accuracy. A conservative scalar transport algorithm is also applied to discretize k - c equations in this model. The eddy viscosity is calculated from the k-c turbulent model. The resulting model is mass and momentum conservative. The model is verified by two examples to simulate unsteady small amplitude free surface flows where non-hydrostatic pressures have a considerable effect on the velocity field, and then applied to simulate the tidal flow in the Bohai Sea.

NUMERICAL MODELLING OF FREE-SURFACE FLOWS WITH BOTTOM AND SURFACE-LAYER PRESSURE TREATMENT

A new non-hydrostatic numerical model with the three-dimensional Navier-Stokes equations on structured grids was constructed and discussed. The algorithm is based upon a staggered finite difference Crank-Nicholson scheme on a Cartesian grid. The eddy viscosity coefficient was calculated by the efficient k-ε turbulence model. A new surface-layer non-hydrostatic treatment and a local cell bottom treatment were introduced so that the three-dimensional model is fully non-hydrostatic and is free of any hydrostatic assumption. The developed model is second-order accuracy in both time and space when semi-implicit coefficient is set to 0.5. The validity of the present solution algorithm was demonstrated from its application to the three-dimension channel flow and the wave propagation over a submerged bar problems.

Effect of Stokes drift on upper ocean mixing

Stokes drift is the main source of vertical vorticity in the ocean mixed layer. In the ways of Coriolis - Stokes forcing and Langmuir circulations, Stokes drift can substantially affect the whole mixed layer. A modified Mellor-Yamada 2. 5 level turbulence closure model is used to parameterize its effect on upper ocean mixing conventionally. Results show that comparing surface heating with wave breaking, Stokes drift plays the most important role in the entire ocean mixed layer, especially in the subsurface layer. As expected, Stokes drift elevates both the dissipation rate and the turbulence energy in the upper ocean mixing. Also, ilffluence of the surface heating, wave breaking and wind speed on Stokes drift is investigated respectively. Research shows that it is significant and important to assessing the Stokes drift into ocean mixed layer studying. The laboratory observations are supporting numerical experiments quantitatively.

Simulation of buoyancy-induced turbulent flow from a hot horizontal jet

Experimental visualizations and numerical simulations of a horizontal hot water jet entering cold water into a rectangular storage tank are described. Three different temperature differences and their corresponding Reynolds numbers are considered. Both experimental visualization and numerical computations are carried out for the same flow and thermal conditions. The realizable k-e model is used for modeling the turbulent flow while the buoyancy is modeled using the Boussinesq approximation. Polynomial approximations of the water properties are used to compare with the Boussinesq approximation. Numerical solutions are obtained for unsteady flow while pressure, velocity, temperature and turbulence distributions inside the water tank as well as the Froude number are analyzed. The experimental visualizations are performed at intervals of five seconds for all different cases. The simulated results are compared with the visualized results, and both of them show the stratification phenomena and buoyancy force effects due to temperature difference and density variation. After certain times, depending on the case condition, the flow tends to reach a steady state.

Numerical Study on Plume Interaction Above An AlternatingDiffuser in Stagnant Water

The plume interaction above an alternating diffuser in stagnant water is studied with 3D Reynolds-averaged NavierStokes equations （RANS） combined with a buoyancy-extended κ-ε model. The steady three-dimensional turbulent flow and temperature fields are computed by use of the finite volume method on a non-uniform high resolution orthogonal grid. The numerical predictions demonstrate a generic flow pattern for different turbulent heated jet discharges： the buoyant jets on each side of the diffuser first merge to form an essentially two-dimensional plume which bends back toward the diffuser centerline due to a low pressure cavity. In general, an under-pressure exists in the cavity until the plumes merge; the pressure increases to slightly positive afterwards. Two-dimensionality of the scalar and flow field is attained much later than the point of zero pressure. The position of merging point is governed by mainly four parameters - the discharge densimetfic Froude number, the port diameter and space, and the horizontal distance between alternating jet nozzles. A formula from numerical simulations is obtained through regression analysis and it is used to predict the position of plume merging point. The predicted temperature fields are comparable to previous experiments.

Numerical simulation of flow past circular duct

The Renormalization Group （RNG） k- ε turbulence model and Volume of Fluid （VOF） method were employed to simulate the flow past a circular duct in order to obtain and analyze hydraulic parameters. According to various upper and bottom gap ratios, the force on the duct was calculated. When the bottom gap ratio is 0, the drag force coefficient, lift force coefficient, and composite force reach their maximum values, and the azimuth reaches its minimum. With an increase of the bottom gap ratio from 0 to 1, the drag force coefficient and composite force decrease sharply, and the lift force coefficient does not decreases so much, but the azimuth increases dramatically. With a continuous increase of the bottom gap ratio from 1 upward, the drag force coefficient, lift force coefficient, composite force, and azimuth vary little. Thus, the bottom gap ratio is the key factor influencing the force on the circular duct. When the bottom gap ratio is less than 1, the upper gap ratio has a remarkable influence on the force of the circular duct. When the bottom gap ratio is greater than 1, the variation of the upper gap ratio has little influence on the force of the circular duct.

Numerical Analysis of Turbulent Sonic Jets from Two-Dimensional Convergent Nozzles

The numerical analysis of a turbulent sonic jet from a two-dimensional convergent nozzle has been carried out using the compressible k-? turbulence model and TVD finite difference scheme. Numerical conditions have been varied over a range of operating pressure ratios from 1.893 to 6.0 which cover the jet flow conditions from correctly expanded to underexpanded. Numerical flow visualization of sonic jet structure using the computer schlieren, a relation between shock cell length in the jet with the operating pressure ratio and the pressure distribution along jet centerline are obtained. Also, a transition process of a two-dimensional sonic jet from correctly expanded to underexpanded conditions is shown in detail and a flow model of jet structure is proposed.

DETACHED EDDY SIMULATION OF HYDRAULIC CHARACTERISTICS ALONG THE SIDE-WALL AFTER A NEW ARRANGEMENT-SCHEME OF THE SUDDEN LATERAL ENLARGEMENT AND THE VERTICAL DROP

In order to study the cavitation damage in a side-wall when a sudden lateral enlargement and a vertical drop are imposed at the radial gate, a new arrangement-scheme is proposed, where the sudden lateral enlargement and the vertical drop can be imposed at the outlet of the gate chamber. The hydraulic characteristics along the side-wall are simulated by the detached eddy simulation and the Volume Of Fluid （VOF） method. The numerical results agree well with those of experiment. The experimental and numerical results show that the flow condition is smooth with only a weak water-wing appearing behind the lateral cavity, and the length of the lateral cavity becomes longer and is mainly affected by the size of the lateral enlargement and the zone of negative pressure after the water impacts the side-wall would disappear. The hydraulic characteristics of the new arrangement-scheme are beneficial to the prevention of the cavitation damage in the side-wall and the Detached Eddy Simulation （DES） with the VOF method can well predict the hydraulic characteristics after the new arrangement-scheme of the sudden lateral enlargement and the vertical drop.

EFFECT OF DISCHARGE RATIO ON FLOW CHARACTERISTICS IN 90°EQUAL-WIDTH OPEN-CHANNEL JUNCTION

For the 90° equal-width open-channel junction flow, the Reynolds averaged Navier-Stokes equations are solved while using the 3-D κ- ω model. The mean flow pattern and the secondary current are obtained. The model is validated by experimental data, and then applied to investigate the effect of the discharge ratio on the shape of separation zone shape, the cross-sectional mean flow angle and the contraction coefficient. The results are fairly close to those of the prior studies. The numerical modeling is both less time-consuming and less expensive to obtain the various flow parameters needed for engineering design.

NUMERICAL SIMULATION OF PROPAGATION AND BREAKING PROCESSES OF A FOCUSED WAVES GROUP

A two-dimensional numerical wave flume is developed to study the focused waves group propagation and the consequent breaking processes. The numerical model is based on the Reynolds-Averaged Navier-Stokes （PANS） equations, with the standard k - c turbulence model to simulate the turbulence effects. To track the complicated and broken free-surface, the Volume Of Fluid （VOF） method is employed. The numerical model combines the ＂Partial Cell Treatment （PCT）＂ method with the ＂Locally Relative Stationary （LRS）＂ concept to treat the moving wave paddle so that various waves can be generated directly in a fixed Cartesian grid system. The theoretical results of the linear and nonlinear waves are used to validate the numerical wave flume firstly, and then a plunging breaking wave created by a focused waves group is simulated. The numerical results are compared to the experimental data and other simulation results, with very good agreements. The turbulence intensity, the flow field and the energy dissipation in the breaking processes are analyzed based on the numerical results. It is shown that the present numerical model is efficient and accurate for studying the waves group generation, the waves packet propagation, and the wave breaking processes.