Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

A proper orthogonal decomposition (POD)‐based reduced‐order model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A space‐marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced‐order model. In order to improve the accuracy and the stability of the reduced‐order model in the presence of a high Reynolds number, we applied a Sobolev H₁ norm calibration to the POD construction process. Finally, some numerical tests with a high‐fidelity model as well as the POD reduced‐order model were carried out to demonstrate the efficiency and the accuracy of the reduced‐order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H₁ norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced‐order PNS models. Copyright © 2011 John Wiley & Sons, Ltd.     

MODELLING OF FLOW IN RECTANGULAR SEDIMENTATION TANKS BY AN EXPLICIT THIRD-ORDER UPWINDING TECHNIQUE

A new numerical model has been developed to simulate the transport of dye in primary sedimentation tanks operating under neutral density conditions. A multidimensional algorithm based on a new skew third-order upwinding scheme (STOUS) is used to eliminate numerical diffusion. This algorithm introduces cross-difference terms to overcome the instability problems of the componentwise one-dimensional formulae for simulating multi-dimensional flows. Small physically unrealistic overshooting and undershooting have been avoided by using a well-established technique known as the universal limiter. A well-known rotating velocity field test was used to show the capability of STOUS in eliminating numerical diffusion. The STOUS results are compared with another third-order upwinding technique known as UTOPIA. The velocity field is obtained by solving the equations of motion in the vorticity–streamfunction formulation. A k– ϵ model is used to simulate the turbulence phenomena. The velocity field compares favourably with previous measurements and with UTOPIA results. An additional differential equation governing the unsteady transport of dye in a steady flow field is solved to calculate the dye concentration and to produce flow-through curves (FTCs) which are used in evaluating the hydraulic efficiency of settling tanks. The resulting FTC was compared with both measurements and numerical results predicted by various discretization schemes. © 1997 by John Wiley & Sons, Ltd.     

A POD-based reduced-order finite difference extrapolating model with fully second-order accuracy for non-stationary Stokes equations

A proper orthogonal decomposition (POD) reduced-order finite difference (FD) extrapolating model is established for the channel flow with local expansion denoted by non-stationary Stokes equations. The POD-based reduced-order numerical model to produce the solutions on the time span [T₀, T] (T₀ ? T) are obtained by extrapolation and iteration from the very short time span [0, T₀] information. The guides to choose the number of POD basis and renew POD basis are provided, and an implementation for solving the POD-based reduced-order FD extrapolating model is given. Some numerical experiments are used to show that the POD-based reduced-order FD extrapolating model is feasible and efficient for simulating the channel flow with local expansion.     

A Virtual Numerical Simulator for Aeration Effects in Lake Eutrophication

In this paper, the eutrophication process in reservoirs is briefly presented, then one of the best remedial actions against it, i.e. aeration, is developed. Multiphase flows are then described to show the complexity of the process and related modeling issues. A first order approximation is then suggested through one-phase flow analysis based on the conventional Navier-Stokes equations to simulate virtually aeration effects in natural reservoirs or lakes by prescribing appropriate internal conditions for velocity to simulate air injection. Though the flow features three-dimensional aspects with variable density, only two-dimensional incompressible fluid flow configurations are considered in this paper to demonstrate the feasibility of the methodology developed. Numerical experiments are then run for scenario analysis. Though the results are somewhat qualitative, they are encouraging and with some kind of validation and calibration a cheap alternative can be offered for studying the aeration process.     

Numerical simulation of turbulent impinging jet on a rotating disk

The calculations of quasi‐three‐dimensional momentum equations were carried out to study the influence of wall rotation on the characteristics of an impinging jet. The pressure coefficient, the mean velocity distributions and the components of Reynolds stress are calculated. The flow is assumed to be steady, incompressible and turbulent. The finite volume scheme is used to solve the continuity equation, momentum equations and k–ε model equations. The flow characteristics were studied by varying rotation speed ω for 0?ω?167.6 rad/s, the distance from nozzle to disk (H/d) was (3, 5, 8 and 10) and the Reynolds number Re base on V_J and d was 1.45 × 10⁴. The results showed that, the radial velocity and turbulence intensity increase by increasing the rotation speed and decrease in the impingement zone as nozzle to disk spacing increases. When the centrifugal force increases, the radial normal stresses and shear stresses increase. The location of maximum radial velocity decreases as the local velocity ratio (α) increases. The pressure coefficient depends on the centrifugal force and it decreases as the distance from nozzle to plate increases. In impingement zone and radial wall jet, the spread of flow increases as the angular velocity decreases The numerical results give good agreement with the experiment data of Minagawa and Obi (Int. J. of Heat and Fluid Flow 2004; 25 :759–766). Copyright © 2006 John Wiley & Sons, Ltd.     

Drag reduction of motor vehicles by active flow control using the Coanda effect

A test facility has been constructed to realistically simulate the flow around a two dimensional car shaped body in a wind tunnel. A moving belt simulator has been employed to generate the relative motion between model and ground. In a first step, the aerodynamic coefficients c _L and c _D of the model are determined using static pressure and force measurements. LDA-measurements behind the model show the large vortex and turbulence structures of the near and far wake. In a second step, the ambient flow around the model is modified by way of an active flow control which uses the Coanda effect, whereby the base-pressure increases by nearly 50% and the total drag can be reduced by 10%. The recirculating region is completely eliminated. The current work reveals the fundamental physical phenomena of the new method by observing the pressure forces on the model surface as well as the time averaged velocities and turbulence distributions for the near and far wake. A theory resting on this empirical information is developed and provides information about the effectiveness of the blowing method. For this, momentum and energy equations were applied to the flow around the vehicle to enable a validation of the theoretical results using experimental values. Received: 9 June 1998 / Accepted: 20 July 1999     

NUMERICAL CALCULATION OF 2D,UNSTEADY FLOW IN CENTRIFUGAL PUMPS: IMPELLER AND VOLUTE INTERACTION

A numerical model is developed for calculating the two-dimensional, unsteady, incompressible and turbulent flow within the rotating impeller and stationary volute of an industrial centrifugal pump. The objective is the investigation and comprehension of the instantaneous behaviour of centrifugal pumps, aiming at the reduction of vibrations, radial forces and hydraulic noise. The computation is performed within a blade-to-blade streamtube for the impeller and a tube normal to the axis of rotation for the volute. The equations to be solved are the unsteady Reynolds-averaged Navier–Stokes equations along with the continuity equation and the unsteady κ–ϵ equations for turbulence modelling. The finite volume method is applied for space discretization and an implicit scheme for time discretization. A multidomain overlapping grid technique is used for matching together the relative flow field calculated within the rotating impeller and the absolute one calculated within the stationary volute. In this way the impeller and volute interaction is directly taken into account. The numerical model is validated for a centrifugal pump of N _q=32 under design flow conditions. Comparisons between calculation and measurements show fairly good agreement.     

The null field approach to two-dimensional elastostatics

The vector basis functions, necessary for solving two-dimensional inclusion problems in an elastic solid under time independent conditions by means of the null field approach (T-matrix method), are obtained as a zero frequency limit of the corresponding basis functions commonly used in elastodynamics. The expansion of the fields appearing in the surface integral representation of the static displacement can thus be achieved, leading to the T-matrix equations of 2d-elastostatics. We specialize the problem to the simple boundary condition case of a single cavity and develop the analytical expressions as much as possible before numerical implementation. A numerical test for the ellipse and some examples for the superellipse, with applied static pressure or shear stress at infinity, are given.     

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T _a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.     

Wall modeling via function enrichment within a high‐order DG method for RANS simulations of incompressible flow

We present a novel approach to wall modeling for the Reynolds‐averaged Navier‐Stokes equations within the discontinuous Galerkin method. Wall functions are not used to prescribe boundary conditions as usual, but they are built into the function space of the numerical method as a local enrichment, in addition to the standard polynomial component. The Galerkin method then automatically finds the optimal solution among all shape functions available. This idea is fully consistent and gives the wall model vast flexibility in separated boundary layers or high adverse pressure gradients. The wall model is implemented in a high‐order discontinuous Galerkin solver for incompressible flow complemented by the Spalart‐Allmaras closure model. As benchmark examples, we present turbulent channel flow starting from Re_τ=180 and up to Re_τ=100000 as well as flow past periodic hills at Reynolds numbers based on the hill height of Re_H=10595 and Re_H=19000.     

On Fokker–Planck Equations for Turbulent Reacting Flows. Part 2. Filter Density Function for Large Eddy Simulation

The application of large eddy simulation (LES) to turbulent reacting flow calculations is faced with several closure problems. Suitable parametrizations for filtered reaction rates for instance are hardly available in general. A way to overcome these problems is investigated here. This is done by extending LES equations for filtered velocities and scalars (mass fractions of species and temperature) to equations that involve subgrid scale (SGS) fluctuations. Such equations are called filter density function (FDF) methods because they determine the FDF, which is essentially the probability density function of SGS variables. The FDF model considered involves only three parameters: C ₀ that controls the generation of velocity fluctuations and two parameters which determine the relaxation of velocity and scalar fluctuations. The consideration of this model may be seen as the analysis of a limiting case: the implications of the most simple equations for the dynamics of SGS fluctuations are investigated in this way. These equations were proved recently by various simulations. Here, the FDF model is used analytically to improve simpler methods. Existing models for the SGS stress tensor in velocity LES equations and the diffusion coefficient in scalar FDF equations are generalized in this way. The advantages of these models compared to existing ones are pointed out. These investigations provide further evidence for the suitability of the FDF model considered and they provide its parameters. A theoretical value C ₀ = 19/12 is derived, which agrees very well with the results of direct numerical simulation. This estimate implies the same value for the universal Kolmogorov constant of the energy spectrum, which is consistent with the results of many measurements. The other two model parameters can be obtained then by dynamic procedures. Therefore, the closure problems of LES equations are overcome in this way such that adjustable parameters are not involved.     

Explicit integration of one problem of motion of the generalized Kowalevski top

In the problem of motion of the Kowalevski top in a double force field the four-dimensional invariant submanifold of the phase space was pointed out by [Kharlamov, M.P., 2002. Mekh. Tverd. Tela 32, 33–38]. We show that the equations of motion on this manifold can be separated by the appropriate change of variables, the new variables s₁, s₂ being elliptic functions of time. The natural phase variables (components of the angular velocity and the direction vectors of the forces with respect to the movable basis) are expressed via s₁, s₂ explicitly in elementary algebraic functions.     

Integrated radial basis functions‐based differential quadrature method and its performance

In this paper, indirect radial basis function networks (IRBFN) proposed by Nam and Tranh (Neural Networks 2001; 14 (2):185–199; Appl. Math. Modelling 2003; 27 :197–220) are incorporated into the differential quadrature (DQ) approximation of derivatives. For simplicity, this new variant of RBF‐DQ approach is named as iRBF‐DQ method. The proposed approach is validated by its application to solve the one‐dimensional Burger's equation, and simulate natural convection in a concentric annulus by solving Navier–Stokes equations. It was found that as compared to the benchmark data, the iRBF‐DQ approach can provide more accurate results than the original RBF‐DQ method. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element technique to solve the elastic strain for Leonov fluid flow

The finite element method is used to find the elastic strain (and thus the stress) for given velocity fields of the Leonov model fluid. With a simple linearization technique and the Galerkin formulation, the quasi-linear coupled first-order hyperbolic differential equations together with a non-linear equality constraint are solved over the entire domain based on a weighted residual scheme. The proposed numerical scheme has yielded efficient and accurate convective integrations for both the planar channel and the diverging radial flows for the Leonov model fluid. Only the strain in the inflow plane is required to be prescribed as the boundary conditions. In application, it can be conveniently incorporated in an existing finite element algorithm to simulate the Leonov viscoelastic fluid flow with more complex geometry in which the velocity field is not known a priori and an iterative procedure is needed.     

Measurements of the first normal-stress difference in a new Rheo-dilatometer for molten polymers: Triple-step-shear-strain tests for all K-BKZ constitutive equations

Following the 3rd of a triple-step-shear-strain sequence, the relaxation of the first normal-stress difference,N ₁ (t), has been measured for a sample (IUPAC A) of low-density polyethylene at 150 °C over a range of shear-strain magnitudes up to 28 shear units. A new cone-plate Rheodilatometer was used, giving sample volume change data as well as thrust and torque data during cone rotation. At 130 °C, prior measurements (described elsewhere) gave an upper limit of 0.001% for the volume change with IUPAC A; this gave an estimated error of only 1.7% in determiningN ₁(t) from time-dependent thrust data. Wagner's network-rupture model (based on a postulated strain-induced irreversible network disentanglement process) gave quantitative agreement withN ₁(t) data for IUPAC A at 150 °C over some of the ranges of time and shear-strain investigated. Wagner's earlier constitutive equation (of K-BKZ type) gave no quantitative agreement. Using a special choice of strain magnitudes in which the 3rd is half that of the 1st and 2nd, it is shown that a critical test of all K-BKZ equations can be made on the basis ofN ₁ (t) data only. This test excludes all K-BKZ equations for IUPAC A under the conditions here used. In all the present investigations, the 1st and 3rd shears had the same sign while the 2nd had the opposite sign.     

曝气速度对曝气池流场影响的数值模拟分析

采用欧拉-欧拉多相流模型,辅以RNG k-ε湍流模型对一简化的推流式曝气池进行了数值模拟,控制方程采用有限体积法离散,并采用PISO(Pressure-Implicit with Splitting of Operators)算法求解.通过模拟得到了不同曝气速度下曝气池特征断面的气相体积分数、气液两相速度的分布规律以及...     

Evaluation of Smagorinsky‐based subgrid‐scale models in a finite‐volume computation

Smagorinsky‐based models are assessed in a turbulent channel flow simulation at Re_b=2800 and Re_b=12500. The Navier–Stokes equations are solved with three different grid resolutions by using a co‐located finite‐volume method. Computations are repeated with Smagorinsky‐based subgrid‐scale models. A traditional Smagorinsky model is implemented with a van Driest damping function. A dynamic model assumes a similarity of the subgrid and the subtest Reynolds stresses and an explicit filtering operation is required. A top‐hat test filter is implemented with a trapezoidal and a Simpson rule. At the low Reynolds number computation none of the tested models improves the results at any grid level compared to the calculations with no model. The effect of the subgrid‐scale model is reduced as the grid is refined. The numerical implementation of the test filter influences on the result. At the higher Reynolds number the subgrid‐scale models stabilize the computation. An analysis of an accurately resolved flow field reveals that the discretization error overwhelms the subgrid term at Re_b=2800 in the most part of the computational domain. Copyright © 2002 John Wiley & Sons, Ltd.