Three-dimensional Viscous Flow Research Articles

Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

We introduce a novel positivity-preserving numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier–Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.

Aerodynamic design of a high-efficiency two-stage axial turbine, using streamline curvature method and performance optimisation by clocking of stator blades

Abstract Aerodynamic design of a high-efficiency two-stage axial turbine is carried out using a hybrid method through implantation of a two-step design procedure. In the first step, the well-known streamline curvature (SLC) and free vortex (FV) methods are properly combined to establish three-dimensional geometries of the blades at each row and to obtain the flow field properties. The second step is provided to obtain the highest aerodynamic efficiency by optimum clocking of the second stator blades relative to the first ones through executing steady and unsteady computational fluid dynamics (CFD) of three-dimensional viscous flow. Slight discrepancies were observed between gas dynamics results of the SLC and those of CFD. Total pressure and temperature at the turbine outlet, obtained from SLC method, differed from those obtained by 3D-CFD technique by 13.06% and 1.88% respectively. Aerodynamic efficiency of the turbine is obtained about 91.83%, based on 3D-CFD. Time-averaged results showed that under the optimum clocking of the second row stator blades, inlet total pressure and output power of the second rotor increase by 0.23%, and 0.93%, respectively, in comparison to the worst clocking case. These augmentations resulted in increased total to total efficiency of the second stage by 0.444%. Additionally, the total output power of the two stages increased by 0.71% through the optimum clocking. Modeling the unsteady wake flow trajectory within the blades passages confirmed that all of these beneficial effects happen if the upstream wake impinges on the leading edge region of the second stator blades.

Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals

Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow.

The acceleration effect of pump as turbine system during starting period

In order to reveal the influence of starting acceleration on starting process of a pump as turbine system, this paper carries out a numerical calculation of the three-dimensional viscous unsteady flow of pump as turbine circulating piping system under three starting acceleration conditions, and obtains the external and internal flow characteristics of each overflow component during the starting process, and also analyzes the energy loss of each component in the piping system in depth with the help of entropy production method and Q criterion method. The results show that during system start-up, the flow rate and outlet static pressure curves of the pump as turbine are hysteresis relative to the rotational speed, the head curve is similar to a linear rise during slow and medium speed start-up, while it shows a parabolic rise during rapid start-up, the entropy production and vorticity in the impeller domain of the pump as turbine are mainly distributed between the blades, and the distribution decreases during start-up. In addition, the pump similarity law does not apply to the performance prediction during the transient start of the pump as turbine.

Characteristic-Based Fluid Flow Modeling between Two Eccentric Cylinders in Laminar and Turbulent Regimes

Determining the flow between eccentric cylinders is crucial in a wide range of industries. The governing equations for the flow between eccentric cylinders cannot be solved analytically. Therefore, three-dimensional incompressible viscous fluid flow between eccentric and concentric cylinders has numerically been simulated in this paper to investigate them using a characteristic-based approach. The first-order characteristic-based scheme is used to calculate convective terms, whereas the second-order averaging technique is used to calculate viscous fluxes. The Taylor number, eccentricity distance, Reynolds number, and radius ratio are considered the controlling parameters of fluid flow between the cylinders. The influence of flow between cylinders on flow patterns is presented in terms of velocity, pressure, and flow contours. It is found that at a constant Taylor number, the asymmetric centrifugal forces produce the Taylor vortices on the right of the internal rotating cylinder as the eccentric distance increases. When the eccentric distance increases, the magnitude of shear stress and its fluctuation on the cylinder wall, as well as the pressure on the cylinder wall, rise. The numerical results obtained were validated by comparing them to previously published experimental results, which showed a high level of agreement.

Numerical and Experimental Analysis of Transient Flow Field and Pressure Pulsations of an Axial-Flow Pump Considering the Pump–Pipeline Interaction

The internal flow in a vertical axial-flow pump is a complex unsteady three-dimensional viscous flow. An unstable flow often produces complex flow phenomena such as flow separation, vortices, and secondary reflux, which reduces the operating efficiency of the pump and can endanger safety and stability. In this paper, computational fluid dynamics is used to calculate the flow characteristics in an axial-flow pump using the shear stress transport and curvature correction (SST-CC) model for turbulence modified to account for the rotational curvature. Furthermore, the dependability of the numerical results was confirmed by a test with an actual model of a pump. The transient deviation angle at the impeller inlet of the pump, the stream field attributes in various spanwise parts of the impeller and guide vane, and the velocity distributions at the impeller inlet and outlet were analyzed. The omega method was utilized to recognize the vortex structure inside the guide vane. Moreover, the development of the transient vortex structure inside the guide vane was studied. As the flow rate increased, the scale and turbulent kinetic energy of the vortex structure gradually decreased. The time-domain graph for the impeller inlet is clearly periodic, with three peaks and three troughs in an impeller rotational period. The dominant frequency in the spectrum at each monitoring point was basically the blade frequency, and the secondary dominant frequency was twice the blade frequency.

On the Cauchy problem of the full Navier-Stokes equations for three-dimensional compressible viscous heat-conducting flows subject to large external potential forces

On the Cauchy problem of the full Navier-Stokes equations for three-dimensional compressible viscous heat-conducting flows subject to large external potential forces

Refined blow-up criteria for the three-dimensional viscous compressible flows with large external potential force and general pressure

Refined blow-up criteria for the three-dimensional viscous compressible flows with large external potential force and general pressure

Vortex Patterns Investigation and Enstrophy Analysis in a Small Scale S-CO2 Axial Turbine

Supercritical carbon dioxide (S-CO2) Brayton cycle system is a promising closed-loop energy conversion system frequently mentioned in the automotive and power generation field in recent years. To develop a suitable design methodology for S-CO2 turbines with better performance, an understanding of the vortex flow patterns and associated aerodynamic loss inside a S-CO2 turbine is essential. In this paper, a hundred-kilowatt level S-CO2 axial turbine is designed and investigated using a three-dimensional transient viscous flow simulation. The NIST Span and Wagner equation of state model that considers the real gas effects is utilized to estimate the thermodynamic properties of the supercritical fluid. The numerical methods are experimentally validated. The results indicates that the aspect ratio and tip-to-hub ratio are different in the S-CO2 turbine from that in the gas turbine, and the vortex flow patterns are influenced notably by these geometrical parameters. Both the vortex structure and moving tracks of passage vortices are changed as a result of large centrifugal force. An interaction between tip leakage vortex and hub passage vortex is observed in the impeller passage and its formation and development mechanism are revealed. To further explore the aerodynamic loss mechanism caused by vortex interaction, the energy loss in the impeller passage is analyzed with the enstrophy dissipation method, which can not only accurately calculate the energy loss but also estimate how the vortical motions occur. It is found that the enstrophy and energy loss can be effectively reduced by vortex interaction between tip leakage vortex and hub passage vortex. The results in this study would increase the knowledge of vortex flow patterns in S-CO2 turbine and the proposed enstrophy production method can be used intuitively to provide a reference for flow vortical motion study in turbines.

Numerical investigation of the onset of three-dimensional characteristics in flow past a pair of square cylinders at various arrangements

An incompressible Navier–Stokes solver has been developed to analyze the three-dimensional viscous flow around a pair of square cylinders at a low Reynolds number (Re = 100) at various arrangements. A semi-implicit finite difference scheme was employed on a staggered grid, which facilitated the easier application of boundary conditions while achieving satisfactory accuracy and stability. A third-order upwind (Kawamura–Kuwahara) scheme was applied to discretize the convective terms and a second-order central difference scheme to discretize the diffusive terms. Time integration was performed using a second-order accurate Adams–Bashforth scheme. An iterative pressure–velocity correction was used to obtain a divergence-free velocity field. Aspects of the flow such as vorticity profiles, the time evolution of lift and drag coefficients, and three-dimensionality appearance were studied. Three-dimensional characteristics ceased to appear in a tandem arrangement, irrespective of the gap ratio. In contrast, the side by side arrangement showed a clear three-dimensional transition in the flow when analyzing the vortex structures. Further investigations carried out for staggered arrangement keeping the center to center distance constant and varying the stagger angle (α) reveal that the three-dimensional behavior starts at α = 30° and keeps progressing with the angle increase.

Application of a New Hybrid Immersed Boundary Method for the Simulation of Viscous Compressible Flows

In this paper, we present a hybrid immersed boundary method (HIBM) for the simulation of two- and three-dimensional compressible viscous flows around stationary and moving obstacles. The proposed approach combines the compressible boundary condition-enforced IBM proposed by Qiu et al. and the Brinkman penalization methods. The boundary condition-enforced IBM uses a fractional step approach alongside a Dirac delta function to satisfy the boundary conditions on the body surface. Although this method works properly in subsonic regimes, it cannot correctly simulate the wake region for a supersonic flow. On the other hand, the penalization method considers the body as a porous media with a low permeability that forces the velocity and energy inside the body to converge to the body velocity and energy. However, unlike the boundary condition-enforced IBM, the streamlines penetrate the body. In the present approach, the positive features of the above-mentioned methods were included and their drawbacks were excluded. The proposed approach was applied using the finite volume method and an E-CUSP scheme. The performance of the proposed method was numerically evaluated in simulating compressible fluid flow around both stationary and moving boundaries, showing a close agreement with other numerical and experimental data available in the literature. Further, the effects of geometry, Reynolds, and Mach numbers were investigated on the supersonic flow field around elliptical cylinders of various aspect ratios. The results revealed that increasing the aspect ratio led to an increase in the shock standoff distance, recirculation zone, and drag coefficient.

Transient Behavior of a Prototype Centrifugal Pump with Abrupt Decreasing Flow Rate

Centrifugal pump often operates at different working flow rates to meet engineering requirement. To better reveal the transient behavior of centrifugal pump in the process of decreasing flow rate, the finite volume method (FVM), RNG k-ε turbulence model, sliding mesh technology, and user-defined functions (UDF) were employed to simulate the three-dimensional unsteady viscous incompressible flow in a low-specific-speed centrifugal pump during the abrupt valve-off period. The results show that the differences are very obvious between transient and quasi-steady calculations. The velocity is maximum on the wall of hub and shroud, while the velocity is minimum and uniform distribution at middle positions. The transient flow field lags behind the quasi-steady flow field, which may be related to the reasons; namely, kinetic energy cannot convert pressure energy in time.

The Influence of Mesh Resolution on 3D RANS Flow Simulations in Turbomachinery Flow Parts

The question of the difference mesh refinement degree influence on the results of calculation of the three-dimensional viscous gas flows in the flow parts of turbomachines using the RANS flow models and second order numerical methods is considered. Calculations of flows for a number of turbine and compressor grids on successively refining grids have been performed. We used H-type grids with approximate orthogonalization of cells in the boundary layer. The calculations were carried out using a CFD solver F with the use of an implicit ENO scheme of the second order, a local time step, and a simplified multigrid algorithm. When calculating the flow on fine grids, the following were used: convergence acceleration tools implemented in the solver; truncation of the computational domain with subsequent distribution of the results based on the symmetry property; the computational domain splitting into parts and computations parallelizing. Comparison of the obtained results is carried out, both in terms of qualitative resolution of the complex structure of three-dimensional flows, and in terms of quantitative assessment of losses. Grid convergence was estimated in two ways. In the first, the characteristic two-dimensional distributions of parameters obtained on different grids were visually compared. The purpose of such comparisons was to evaluate the sufficient degree of solution of both the general structure of the flow in grids and its features, namely, shock waves, contact discontinuities, separation zones, wakes, etc. The second estimation method is based on the grid convergence index (GCI). The GCI calculated from the three-dimensional density field was considered in this paper. It is concluded that for scientific research requiring high accuracy of calculations and detailing of the structure of a three-dimensional flow, very fine difference meshes with the number of cells from 106 to 108 in one blade-to-blade channel are needed, while for engineering calculations, under certain conditions, it is sufficient to use meshes with the number of cells less than 1 million in one blade-to-blade channel.

The universal steady lift and drag theory and the physical origin of lift

Since the birth of modern aerodynamics, various theories on lift and drag have been developed and validated extensively in aeronautical applications. However, the far-field force theory had long remained at low-speed incompressible flow. Based on the analytical solutions of the linearized Navier-Stokes equations in the steady far field, the authors and their collaborators extended the classic Kutta-Joukowski lift theorem to both two- and three-dimensional viscous and compressible flows, and thus filled the long-standing gap in theoretical aerodynamics. Why can the simple formulas based on linearized approximation still be accurately valid for highly nonlinear complex flows? This issue of great interest involves the methodological characteristics and physical mechanism behind the unified force theory and is the first task of this article. Moreover, there has been an abnormal phenomenon regarding the physical origin of lift that, despite the already mature and fully verified rigorous lift theory, various different hypotheses still keep surfacing frequently in various nonscientific publications and media. This indicates that the issue is really complicated and has not been thoroughly clarified in textbooks, monographs, and classrooms around the world. Now, the universality and high conciseness of the unified theory enable one to reach a clear answer to this issue by rigorous logical arguments in the most direct way. This is the second task of this article.

Three-dimensional viscous wake flow in fish swimming - A CFD study

In the present paper, the assessment of the hydrodynamics of fish swimming wake-flow is carried out to identify the main features related to the propulsion performance. Two different fish models with caragiform kinematics are studied: tuna fish (Thunnus atlanticus) and lambari (Astyanax bimaculatus). CFD methods were used to simulate laminar and turbulent flow regimes and a dynamically adaptive mesh discretization (ALE) was employed to reproduce fish motion. Simulations were run in a typical range of Reynolds (Re) and Strouhal (St) numbers of the fish swimming flow investigating hydrodynamics forces and the flow patterns of the induced fish-swimming wake. Firstly, equilibrium condition (drag balancing thrust) for Strouhal was achieved for each Reynolds number and three-dimensional wake structures were evaluated concluding that fish swimming experiencing a forward locomotion movement with a reversal von-Kármán vortex street and a high-velocity jet through the downstream direction. Finally, vorticity dynamics analysis showing that the leading-edge vortex produced by the caudal fin is associated to the peak of thrust in the motion cycle.

Non-modal stability analysis in viscous fluid flows with slippery walls

A study of optimal temporal and spatial disturbance growths is carried out for three-dimensional viscous incompressible fluid flows with slippery walls. The non-modal temporal stability analysis is performed under the framework of normal velocity and normal vorticity formulations. A Chebyshev spectral collocation method is used to solve the governing equations numerically. For a free surface flow over a slippery inclined plane, the maximum temporal energy amplification intensifies with the effect of wall slip for the spanwise perturbation, but it attenuates with the wall slip when perturbation considers both streamwise and spanwise wavenumbers. It is found that the boundary for the regime of transient growth appears far ahead of the boundary for the regime of exponential growth, which raises a question on the critical Reynolds number for the shear mode predicted from the eigenvalue analysis. Furthermore, the eigenvalue analysis or the modal stability analysis reveals that the unstable region for the shear mode decays rapidly in the presence of wall slip, which is followed by the successive amplification of the critical Reynolds number for the shear mode and ensures the stabilizing effect of slip length on the shear mode. On the other hand, for a channel flow with slippery bounding walls, the maximum spatial energy amplification intensifies with the effect of wall slip in the absence of angular frequency, but it reduces with the wall slip if the angular frequency is present in the disturbance. Furthermore, the maximum spatial energy disturbance growth can be achieved if the disturbance excludes the angular frequency. Furthermore, it is observed that the angular frequency plays an essential role in the pattern formation of optimal response. In addition, the pseudo-resonance phenomenon occurs due to external temporal and spatially harmonic forcings, where the pseudo-resonance peak is much higher than the resonance peak.

A three-dimensional gas-kinetic flux solver for simulation of viscous flows with explicit formulations of conservative variables and numerical flux

A truly three-dimensional (3D) gas-kinetic flux solver for simulation of incompressible and compressible viscous flows is presented in this work. By local reconstruction of continuous Boltzmann equation, the inviscid and viscous fluxes across the cell interface are evaluated simultaneously in the solver. Different from conventional gas-kinetic scheme, in the present work, the distribution function at cell interface is computed in a straightforward way. As an extension of our previous work (Sun et al., Journal of Computational Physics, 300 (2015) 492–519), the non-equilibrium distribution function is calculated by the difference of equilibrium distribution functions between the cell interface and its surrounding points. As a result, the distribution function at cell interface can be simply calculated and the formulations for computing the conservative flow variables and fluxes can be given explicitly. To validate the proposed flux solver, several incompressible and compressible viscous flows are simulated. Numerical results show that the current scheme can provide accurate numerical results for three-dimensional incompressible and compressible viscous flows.

Numerical approach of variable thermophysical features of dissipated viscous nanofluid comprising gyrotactic micro-organisms

This article addresses the heat and mass transport phenomena by performing a theoretical analysis of three-dimensional viscous fluid flow containing gyrotactic micro-organisms over a nonlinear stretched surface. Variable magnetic field is considered normal to the stretched surface to control the fluid flow. Thermal transportation is discussed in view of variable thermal conductivity. Variable characteristics of mass diffusion along with chemical reaction are incorporated in mass transportation. Darcy–Forchheimer expression is used to characterise the porous medium. Also, Brownian motion and thermophoresis are incorporated to enhance the diffusion. The governing partial differential equations (PDEs) are derived using boundary layer analysis by assuming small magnetic Reynolds number. Appropriate transformation is used to convert complex system of coupled PDEs into nonlinear ordinary differential equations (ODEs). Transformed problem is then tackled analytically using optimal homotopic procedure. Reliability of the suggested scheme is presented through error reduction table and also by comparing the obtained solution with the published ones. Graphs and tables are prepared to observe the impact of parameters on physical variables. Dimensionless stresses and rate of heat transfer are computed numerically. It has been observed that larger values of Brownian diffusion and thermophoresis increase the fluid temperature. Moreover, dimensionless stresses and rate of heat transfer are computed to check the reliability of the proposed procedure. These values are clearly in an excellent agreement with the previous findings reported in literature.

Rezul'taty chislennogo modelirovaniya vyazkogo potoka v maloraskhodnykh stupenyakh tsentrobezhnykh kompressorov kak osnova sozdaniya matematicheskoy modeli napora

The paper envisages the results of calculation of a threedimensional viscous flow in 18 lowflow stages of centrifugal compressors in the range of optimal conditional flow coefficient Фopt = 0.007-0.018. Gasdynamic characteristics of the internal and polytropic pressure coefficients in full parameters were plotted and compared with experimental data. The calculated gasdynamic characteristics of viscous flow satisfying the engineering accuracy of calculations and experimental character of the curves were obtained using Ansys CFX software

A Novel Vortex Identification Technique Applied to the 3D Flow Field of a High-Pressure Turbine

Abstract The efficiency of modern axial turbomachinery is strongly driven by the secondary flows within the vane or blade passages. The secondary flows are characterized by a complex pattern of vortical structures that origin, interact, and dissipate along the turbine gas path. The endwall flows are responsible for the generation of a significant part of the overall turbine loss because of the dissipation of secondary kinetic energy and mixing out of nonuniform momentum flows. The understanding and analysis of secondary flows requires a reliable vortex identification technique to predict and analyze the impact of specific turbine designs on the turbine performance. However, the literature shows a remarkable lack of general methods to detect vortices and to determine the location of their cores and to quantify their strength. This paper presents a novel technique for the identification of vortical structures in a general 3D flow field. The method operates on the local flow field, and it is based on a triple decomposition of motion proposed by Kolář. In contrast to a decomposition of velocity gradient into the strain and vorticity tensors, this method considers a third, pure shear component. The subtraction of the pure shear tensor from the velocity gradient remedies the inherent flaw of vorticity-based techniques, which cannot distinguish between rigid rotation and shear. The triple decomposition of motion serves to obtain a 3D field of residual vorticity whose magnitude is used to define vortex regions. The present method allows to locate automatically the core of each vortex, to quantify its strength, and to determine the vortex bounding surface. The output may be used to visualize the turbine vortical structures for the purpose of interpreting the complex three-dimensional viscous flow field and to highlight any case-to-case variations by quantifying the vortex strength and location. The vortex identification method is applied to a high-pressure turbine with three optimized blade tip geometries. The 3D flow field is obtained by computational fluid dynamics (CFD) computations performed with Numeca FINE/Open. The computational model uses steady-state Reynolds-averaged Navier–Stokes (RANS) equations closed by the Spalart-Allmaras turbulence model. Although developed for turbomachinery applications, the vortex identification method proposed in this work is of general applicability to any three-dimensional flow field.