Steady Flow Simulations Research Articles

Numerical simulations of steady and unsteady two-phase flows using a homogeneous model

The capability of a homogeneous model to simulate steady and unsteady two-phase flows is investigated. The latter is based on the Euler set of equations supplemented by a complex equation of state describing the thermodynamical behavior of the mixture. No equilibrium assumption is made except for the kinematic equilibrium. The return to the thermodynamical equilibrium is ensured by three source terms that comply with the second law of thermodynamics. The numerical code built on the basis of this model has been verified and some validation results are discussed here. The speed of propagation of a pressure signal is first studied and compared with experimental measurements. Then a more complex situation is investigated: SUPERCANON experiment which corresponds to a sudden depressurization of heated water (associated to a Loss Of Coolant Accident, or LOCA). At last, the results of a numerical experiment of heating of flowing water in a pipe are compared to those obtained with an industrial code.

Well balancing of the SWE schemes for moving-water steady flows

In this work, the exact reproduction of a moving-water steady flow via the numerical solution of the one-dimensional shallow water equations is studied.A new scheme based on a modified version of the HLLEM approximate Riemann solver (Dumbser and Balsara (2016) [18]) that exactly preserves the total head and the discharge in the simulation of smooth steady flows and that correctly dissipates mechanical energy in the presence of hydraulic jumps is presented. This model is compared with a selected set of schemes from the literature, including models that exactly preserve quiescent flows and models that exactly preserve moving-water steady flows.The comparison highlights the strengths and weaknesses of the different approaches. In particular, the results show that the increase in accuracy in the steady state reproduction is counterbalanced by a reduced robustness and numerical efficiency of the models. Some solutions to reduce these drawbacks, at the cost of increased algorithm complexity, are presented.

Drag and inertia coefficients for a circular cylinder in steady plus low-amplitude oscillatory flows

Computer simulations of steady plus low-amplitude oscillatory flow about a circular cylinder are reported at a fixed Reynolds number of 150 based on the steady component. The conventional Keleugan–Carpenter number based on the oscillatory component is fixed at π/5. The oscillation frequency is varied so as to study a wide spectrum of flows where inertial forces dominate at one end and viscous drag forces at the other as a function of the modified Keleugan–Carpenter number. The hydrodynamic force on the cylinder in-line with the flow direction is represented by Morison's equation and an extended version with three terms. The drag and inertia coefficients in Morison's equation are determined by least-squares fits to data directly computed from integration of skin friction and pressure distributions around the periphery of the cylinder. The root-mean-square value of the residue of reconstructed minus directly-computed forces varies between 2 and 41% depending on the flow parameters. Comparable results can be obtained with a semi-theoretical approach using inviscid inertia and quasi-steady viscous drag terms. Physical explanations for the variation of the force coefficients are provided and implications for pertinent flow–structure interactions are discussed.

Prediction of Darcy Permeability and Forchheimer’s Coefficient of Porous Structures Relevant to Regenerator of a Stirling Cryocooler Using a Correlation Based Method

A regenerative heat exchanger is the most vital component in the design of a Stirling cryocooler. Computational Fluid Dynamics (CFD) is the best technique for the design and prediction of the performance of a regenerator. The reliability of the simulation results depend on the accuracy of the Darcy permeability [Formula: see text] and Forchheimer’s inertial coefficient [Formula: see text] used for modeling the momentum transfer in porous media. Usually these coefficients are calculated from pressure drop data obtained from experiment. Because of the requirement of sophisticated equipments for the measurement and analysis of data, experimental study becomes expensive. This paper proposes a friction factor correlation-based method for the prediction of directional permeability and Forchheimer’s inertial coefficient of wire mesh structures relevant to Stirling cryocooler. The friction factor for the flow of helium through #325, #400 and #635 SS wire matrices with porosities of 0.6969, 0.6969 and 0.6312 are calculated using standard correlations and compared with the friction factor given by Clearman et al. based on steady flow experimental study. The friction factor obtained from Blass and Tong/London correlations are in agreement with that of Clearman et al. The viscous and inertial resistances are calculated from the friction factor obtained from Blass and Tong/London correlations. Using these values, the regenerator was modeled as a porous media in Fluent. From the steady flow simulation, pressure drop at different mass flow rates (0.08–1.44[Formula: see text]g/s) is obtained. The maximum deviation of predicted pressure drop from the reported experimental data is 15.14%. The Darcy permeability [Formula: see text] and Forchheimer’s inertial coefficient [Formula: see text] obtained from correlation-based method was used for modeling the oscillatory flow of helium through a #400 regenerator. The pressure amplitude and phase at regenerator exit were obtained at different frequencies. The average deviation of predicted pressure amplitude from the experimental data is 15.83%. The model could predict the phase angle also accurately. Therefore, the proposed method can be used to calculate the hydrodynamic parameters of woven wire screen matrices applied to Stirling cryocoolers.

Lattice Boltzmann Simulation of Steady Flow in a Semi-Elliptical Cavity

AbstractThe lattice Boltzmann method is employed to simulate the steady flow in a two-dimensional lid-driven semi-elliptical cavity. Reynolds number (Re) and vertical-to-horizontal semi-axis ratio (D) are in the range of 500-5000 and 0.1-4, respectively. The effects of Re and D on the vortex structure and pressure field are investigated, and the evolutionary features of the vortex structure with Re and D are analyzed in detail. Simulation results show that the vortex structure and its evolutionary features significantly depend on Re and D. The steady flow is characterized by one vortex in the semi-elliptical cavity when both Re and D are small. As Re increases, the appearance of the vortex structure becomes more complex. When D is less than 1, increasing D makes the large vortexes more round, and the evolution of the vortexes with D becomes more complex with increasing Re. When D is greater than 1, the steady flow consists of a series of large vortexes which superimpose on each other. As Re and D increase, the number of the large vortexes increases. Additionally, a small vortex in the upper-left corner of the semi-elliptical cavity appears at a large Re and its size increases slowly as Re increases. The highest pressures appear in the upper-right corner and the pressure changes drastically in the upper-right region of the cavity. The total pressure differences in the semi-elliptical cavity with a fixed D decrease with increasing Re. In the region of themain vortex, the pressure contours nearly coincide with the streamlines, especially for the cavity flow with a large Re.

Numerical Prediction of Natural Convection Flow in the Presence of Weak Magnetic Prandtl Number and Strong Magnetic Field with Algebraic Decay in Mainstream Velocity

AbstractIn present work, we investigate numerical simulation of steady natural convection flow in the presence of weak magnetic Prandtl number and strong magnetic field by involving algebraic decay in mainstream velocity. Before passing to the numerical simulation, we formulate the set of boundary layer equations with the inclusion of the effects of algebraic decay velocity, aligned magnetic field and buoyant body force in the momentum equation. Later, finite difference method with primitive variable formulation is employed in the physical domain to compute the numerical solutions of the flow field. Graphical results for the velocity, temperature and transverse component of magnetic field as well as surface friction, rate of heat transfer and current density are presented and discussed. It is pertinent to mention that the simulation is performed for different values of algebraic decay parameter α, Prandtl number Pr, magnetic Prandtl number Pm and magnetic force parameter S.

Flood analysis of Wainganga River by using HEC-RAS model

Bhandara city lies on the bank of Wainganga River which is flood prone area. The floods of Wainganga River affects the normal life in the city very frequently. In the present study, USACE Hydrologic Engineering Center's River Analysis System (HEC-RAS 5.0.3) Model is used for the analysis of the flood. The water surface elevation are computed for various flood discharges and return period (25 years, 50 years and 100 years). Total 19 cross sections along the river near the city are considered for the steady and unsteady flow simulations. The flow hydrograph which is generated from previous available data is used for unsteady flow simulation. The output from a HEC-RAS model shows that at cross section 2, 3 for steady flow conditions and cross section 19 for unsteady flow conditions are critical and subjected to the flooding of the adjoining area disturbing the normal life, for reducing the losses the protective measures needs to be undertaken.

Study on Non-Newtonian Behaviors of Lennard-Jones Fluids via Molecular Dynamics Simulations

Using nonequilibrium molecular dynamics simulations, we study the non-Newtonian rheological behaviors of a monoatomic fluid governed by the Lennard-Jones potential. Both steady Couette and oscillatory shear flows are investigated. Shear thinning and normal stress effects are observed in the steady Couette flow simulations. The radial distribution function is calculated at different shear rates to exhibit the change of the microscopic structure of molecules due to shear. We observe that for a larger shear rate the repulsion between molecules is more powerful while the attraction is weaker, and the above phenomena can also be confirmed by the analyses of the potential energy. By applying an oscillatory shear to the system, several findings are worth mentioning here: First, the phase difference between the shear stress and shear rate increases with the frequency. Second, the real part of complex viscosity first increases and then decreases while the imaginary part tends to increase monotonically, which results in the increase of the proportion of the imaginary part to the real part with the increasing frequency. Third, the ratio of the elastic modulus to the viscous modulus also increases with the frequency. These phenomena all indicate the appearance of viscoelasticity and the domination of elasticity over viscosity at high oscillation frequency for Lennard-Jones fluids.

Steady flow and dynamic analyses comparison of an air intake breathing capacity

Analysis of intake breathing capacity is important in order to determine the performance output for any respective engine. Flow coefficient and discharge coefficient are the common parameters used to define the breathing capability and different analysis have been applied by researchers to determine these parameters which include experiments and Computational Fluid Dynamics (CFD) analyses. This study aims to investigate the difference in breathing parameters obtained from steady flow simulation and dynamics simulation. Proton CAMPRO 1.6-litre engine was selected as the reference engine in this study. The experiment involved was the flow bench test, while CFD simulation carried out were the port flow analysis (steady flow) and the cold flow analysis (dynamics). Results obtained indicate that flow coefficient and discharge coefficient from cold flow simulation are always lower than both parameter values obtained from port flow simulation and experiment with large deviations of minimum 15.6% and maximum 27%. Meanwhile, the breathing parameter values from port flow simulation were very close to the experimental data with the minimum deviation of 1.6%. This study concludes that port flow simulation is very accurate for the analysis of defining intake breathing capacity, meanwhile cold flow simulation can be used to predict the trend and lower limit of these parameters especially at low valve lift.

Patterns of steady axisymmetric supersonic compression flows with a Mach disk

Results of a numerical simulation of steady axisymmetric supersonic flows in convergent conical ducts and in overexpanded jets are presented. The characteristic feature of these compression flows is the formation of an initial longitudinally curved shock wave with intensity increasing downstream and toward the flow axis, which is finalized by the generation of a central Mach disk. Computations have demonstrated patterns of an irregular interaction of these shocks followed by the formation of a triple-shock configuration, including a reflected shock and a shear layer with entropy varying across the layer. The formation of triple-shock configurations is analogous to the configurations known for the steady inviscid two-dimensional flows where the irregular reflection of a wedge-generated shock from a wall with Mach stem formation occurs. Either a single triple-shock Mach configuration occurs or a triple-shock configuration corresponding to the von Neumann paradox condition is formed at the considered flow Mach numbers and initial angles of deflection to the axis of the flow behind the longitudinally curved shock wave.

Non-Linear Harmonic flow simulations of a High-Head Francis Turbine test case

This work investigates the use of the non-linear harmonic (NLH) method for a high- head Francis turbine, the Francis99 workshop test case. The NLH method relies on a Fourier decomposition of the unsteady flow components in harmonics of Blade Passing Frequencies (BPF), which are the fundamentals of the periodic disturbances generated by the adjacent blade rows. The unsteady flow solution is obtained by marching in pseudo-time to a steady-state solution of the transport equations associated with the time-mean, the BPFs and their harmonics. Thanks to this transposition into frequency domain, meshing only one blade channel is sufficient, like for a steady flow simulation. Notable benefits in terms of computing costs and engineering time can therefore be obtained compared to classical time marching approach using sliding grid techniques. The method has been applied for three operating points of the Francis99 workshop high-head Francis turbine. Steady and NLH flow simulations have been carried out for these configurations. Impact of the grid size and near-wall refinement is analysed on all operating points for steady simulations and for Best Efficiency Point (BEP) for NLH simulations. Then, NLH results for a selected grid size are compared for the three different operating points, reproducing the tendencies observed in the experiment.

Numerical Simulation of Steady Flow and Forced Convection Heat Transfer from Two Offset Square Cylinders Placed in a Channel

Flow over two isothermal offset square cylinders in a channel is simulated for various offset distance (OD) and Reynolds number (Re). The distance between the cylinders in normal direction and the blockage ratio are fixed. Simulation is carried out for Re that varies from 1 to 40, and OD that varies from 1 to 10 for a fixed Prandtl number equals to 0.71. The hydrodynamic results and forced convection heat transfer are more pronounced for variation in Re and OD. Flow separation begins when Re is >5 from the rear wall of the cylinders. The onset of vortex shedding is found when Re is greater than 45 for OD upto 10. The average skin friction coefficient is reported to reveal the role of Re and OD. The top wall Nusselt number shows higher than remaining walls. The downstream offset cylinder is more pronounced for Re and OD, and it shows a nonlinear trend for average Nusselt number.

An implicit block LU-SGS finite-volume lattice-Boltzmann scheme for steady flows on arbitrary unstructured meshes

This work proposes a fully implicit lattice Boltzmann (LB) scheme based on finite-volume (FV) discretization on arbitrary unstructured meshes. The linear system derived from the finite-volume lattice Boltzmann equation (LBE) is solved by the block lower-upper (BLU) symmetric-Gauss-Seidel (SGS) algorithm. The proposed implicit FV-LB scheme is efficient and robust, and has a low-storage requirement. The effectiveness and efficiency of the proposed implicit FV-LB scheme are validated and verified by the simulations of three test cases in two dimensions: (a) the laminar Blasius flow over a flat plate with Re=105; (b) the steady viscous flow past a circular cylinder with Re=10, 20, and 40; and (c) the inviscid flow past a circular cylinder. The proposed implicit FV-LB scheme is shown to be not only effective and efficient for simulations of steady viscous flows, but also robust and efficient for simulations of inviscid flows in particular.

An Efficient Unsteady Adjoint Optimization System for Multistage Turbomachinery

Unsteady blade row interactions play an important role in the performance of multistage turbomachinery. However, most aerodynamic optimizations of multistage turbomachinery are based on mixing-plane steady flow simulations. To take into account the unsteady flow features in the optimization cycle, this paper develops an adjoint-based unsteady aerodynamic optimization system for multistage turbomachinery. To the authors' best knowledge, this is the first work in the literature conducting the unsteady adjoint aerodynamic optimization of multistage turbomachinery. The unsteady flow equations and the discrete adjoint equations are solved using a finite volume code, with the harmonic balance method adopted to reduce the cost of unsteady simulations. The system is applied to the unsteady aerodynamic optimization of a 1.5-stage compressor. Results show the efficiency and capability of the proposed framework for the unsteady aerodynamic optimization of multistage turbomachinery.

Numerical simulation of steady and unsteady flow for generalized Newtonian fluids

This work presents the numerical solution of laminar incompressible viscous flow in a three dimensional branching channel with circle cross section for generalized Newtonian fluids. The governing system of equations is based on the system of balance laws for mass and momentum. Numerical solution is based on cetral finite volume method using explicit Runge- Kutta time integration. In the case of unsteady computation artificial compressibility method is considered.

Multidisciplinary simulation acceleration using multiple shared memory graphical processing units

In this article, we describe the strategies and programming techniques used in porting a multidisciplinary fluid/thermal interaction procedure to graphical processing units (GPUs). We discuss the strategies for selecting which disciplines or routines are chosen for use on GPUs rather than CPUs. In addition, we describe the programming techniques including use of Compute Unified Device Architecture (CUDA), mixed-language (Fortran/C/CUDA) usage, Fortran/C memory mapping of arrays, and GPU optimization. We solve all equations using the multi-block, structured grid, finite volume numerical technique, with the dual time-step scheme used for unsteady simulations. Our numerical solver code targets CUDA-capable GPUs produced by NVIDIA. We use NVIDIA Tesla C2050/C2070 GPUs based on the Fermi architecture and compare our resulting performance against Intel Xeon X5690 CPUs. Individual solver routines converted to CUDA typically run about 10 times faster on a GPU for sufficiently dense computational grids. We used a conjugate cylinder computational grid and ran a turbulent steady flow simulation using four increasingly dense computational grids. Our densest computational grid is divided into 13 blocks each containing 1033×1033 grid points, for a total of 13.87 million grid points or 1.07 million grid points per domain block. Comparing the performance of eight GPUs to that of eight CPUs, we obtain an overall speedup of about 6.0 when using our densest computational grid. This amounts to an 8-GPU simulation running about 39.5 times faster than running than a single-CPU simulation.

Steady and unsteady mixed convection flow in a cubical open cavity with the bottom wall heated

In this study we analyze experiments and numerical simulations of steady and unsteady mixed convection flow in a cubical cavity located at the bottom of a square channel. The Reynolds numbers based on the mean flow velocity and the channel width are in the range 100⩽Re⩽1500 and the Richardson numbers vary within 0.1⩽Ri⩽10. Particle Image Velocimetry has been used for the measurements in a water channel. Three-dimensional direct numerical simulations have been carried out with a second order finite volume code considering the Boussinesq approximation since, for the experimental conditions considered, the variation of the physical properties with temperature has no significant influence on the overall flow topology. For 100⩽Re⩽1500 and Ri⩽0.1 the flow is steady and it consists in a single roll that exhibits larger velocities as the Richardson number is increased. An unsteady periodic flow is found at Re=100 and Ri=10. Alternate flow ejections from the cavity to the channel occur near the lateral walls while the flow enters the cavity from the channel through the central part of the cavity. A conditional sampling technique has been used to elucidate the evolution of the mean unsteady turbulent flow at Ri=10. It has been found that the alternate flow ejections persist for all the Reynolds analyzed. The computed Nusselt numbers are in general agreement with a previously reported correlation, valid for two dimensional cavities of different aspects ratios.

Pseudo-turbulent heat flux and average gas–phase conduction during gas–solid heat transfer: flow past random fixed particle assemblies

Fluctuations in the gas-phase velocity can contribute significantly to the total gas-phase kinetic energy even in laminar gas–solid flows as shown by Mehrabadi et al. (J. Fluid Mech., vol. 770, 2015, pp. 210–246), and these pseudo-turbulent fluctuations can also enhance heat transfer in gas–solid flow. In this work, the pseudo-turbulent heat flux arising from temperature–velocity covariance, and average fluid-phase conduction during convective heat transfer in a gas–solid flow are quantified and modelled over a wide range of mean slip Reynolds number and solid volume fraction using particle-resolved direct numerical simulations (PR-DNS) of steady flow through a random assembly of fixed isothermal monodisperse spherical particles. A thermal self-similarity condition on the local excess temperature developed by Tenneti et al. (Intl J. Heat Mass Transfer, vol. 58, 2013, pp. 471–479) is used to guarantee thermally fully developed flow. The average gas–solid heat transfer rate for this flow has been reported elsewhere by Sun et al. (Intl J. Heat Mass Transfer, vol. 86, 2015, pp. 898–913). Although the mean velocity field is homogeneous, the mean temperature field in this thermally fully developed flow is inhomogeneous in the streamwise coordinate. An exponential decay model for the average bulk fluid temperature is proposed. The pseudo-turbulent heat flux that is usually neglected in two-fluid models of the average fluid temperature equation is computed using PR-DNS data. It is found that the transport term in the average fluid temperature equation corresponding to the pseudo-turbulent heat flux is significant when compared to the average gas–solid heat transfer over a significant range of solid volume fraction and mean slip Reynolds number that was simulated. For this flow set-up a gradient-diffusion model for the pseudo-turbulent heat flux is found to perform well. The Péclet number dependence of the effective thermal diffusivity implied by this model is explained using a scaling analysis. Axial conduction in the fluid phase, which is often neglected in existing one-dimensional models, is also quantified. As expected, it is found to be important only for low Péclet number flows. Using the exponential decay model for the average bulk fluid temperature, a model for average axial conduction is developed that verifies standard assumptions in the literature. These models can be used in two-fluid simulations of heat transfer in fixed beds. A budget analysis of the mean fluid temperature equation provides insight into the variation of the relative magnitude of the various terms over the parameter space.

Comparison of Steady and Unsteady Flows in a Transonic Radial Vaned Diffuser

Boundary layer suction can be effective in delaying compressor surge, if the surge is triggered by flow separation on the shroud- or hub-casing. This work aims at positioning a suction slot in a radial vaned diffuser, which is thought to be the limiting component in a centrifugal compressor, such as the one considered here. The location of the slot is determined based on the results of both steady and unsteady flow simulations of a transonic centrifugal compressor of a turboshaft. Although the overall performance of the compressor is well-described by steady RANS, large discrepancies are observed between the steady and unsteady simulations of the diffuser flow, discrepancies imply different flow-separation scenarios. Steady results show more low-momentum fluid near the hub, whereas it is concentrated near the shroud in the unsteady simulations, hence no valid physical conclusions can be expected from the steady simulations. Analysis of the instantaneous skin-friction distribution from the unsteady simulations reveals that the separation is fixed and leads to a slot location on the shroud casing, near the diffuser main-vane suction side, so that it covers the range of separation saddle positions as the operating point is changed.

Discrete unified gas kinetic scheme with a force term for incompressible fluid flows

The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which combines the advantages of both lattice Boltzmann equation (LBE) and unified gas kinetic scheme (UGKS), including the simplified flux evaluation scheme, flexible mesh adaption and the asymptotic preserving properties. However, similar to standard LBE, the DUGKS can also be considered as a compressible scheme, and the compressible effect may bring some undesirable errors when it is used to investigate incompressible fluid flows. To eliminate the compressible effect, in this work a new DUGKS with a force term is developed through modifying the equilibrium distribution function. And simultaneously, the non-equilibrium extrapolation (NEE) scheme is also introduced to treat the velocity and pressure boundary conditions. To illustrate the capacity of the present DUGKS, we first performed some numerical simulations of two-dimensional steady and unsteady flows, and conducted a comparison between the present DUGKS and the original one. The results indicate that the present DUGKS can reduce the compressible effect efficiently, and the NEE scheme is also consistent with the second-order accuracy of DUGKS. We then extended the present DUGKS to study the three-dimensional lid-driven flows (LDF) in cubic and deep cavities, and found that the present results are in good agreement with available benchmark results, which indicates the present DUGKS is also accurate and efficient in the study of three-dimensional problems. At last, the structures of vortex in the cubic and deep cavities are also considered, and the symmetric affiliated vortices aside the secondary vortex at Re≥600 can be observed in the deep LDF.