Steady Navier Research Articles

Intrusive and non-intrusive chaos approximation for a two-dimensional steady state Navier–Stokes system with random forcing

Intrusive and non-intrusive chaos approximation for a two-dimensional steady state Navier–Stokes system with random forcing

Numerical simulations of wind loading on the floating photovoltaic systems

This study analyses the fluid dynamics of wind loadings on the floating photovoltaic (PV) system using computational fluid dynamics. The two representative models of pontoon-type and a frame-type with a panel angle of 15° to the ground were investigated. The simulation was performed using the steady solver and incompressible Reynolds-Averaged Navier–Stokes equations with a shear stress transport $$k$$ – $$\omega$$ turbulence model. Inlet condition was 45 m/s steady wind, and outlet condition was set to atmospheric pressure. The results confirmed that wind blowing from the backside of floating PV systems increases drag, lift, and pressure on the first row of the PV panels. The maximum drag and lift coefficient of frame-type PV panels were 0.85 and 0.79, respectively, while that of pontoon-type were 0.81 and 0.65, respectively. The maximum drag and lift coefficient of pontoon-type PV panels with a floating body are 0.29 and 0.25, respectively. Adding the floating body reduced the wind loadings by 70%. Additionally, the blocking back space of the PV panel effectively reduced the maximum drag and lift by 75%, as it prevented the flow impingement.

Airfoil leading edge blowing to control bow shock waves

This manuscript presents a detailed characterization of active control of bow shock waves via leading edge injection, including subsonic coolant ejection and the appearance of Coanda effects. The flow phenomena occurring at steady and pulsating flow injection regimes were analyzed using steady and unsteady two-dimensional Reynolds-Averaged Navier Stokes, leading to a precise evaluation of the thermal load and drag reductions. Steady supersonic injection yields the largest abatement in thermal load and aerodynamic drag, while subsonic or fluctuating ones can also provide significant improvements at reduced cooling mass flow rates. Furthermore, a Coanda effect, causing a non-symmetric flow topology, was observed and analyzed for reduced injection port size. This Coanda effect is due to the sudden expansion happening from the injection port to the main flow and it causes the flow topology at the leading edge to become non-symmetric despite the complete symmetry of the problem. This is the first time in the literature such a phenomenon is documented for a supersonic airfoil leading edge injection. Furthermore, it enables the design of novel flow control strategies for the leading edge shock topology and flow structures in supersonic flows.

Characterization of anisotropic turbulence behavior in pulsatile blood flow

Turbulent-like hemodynamics with prominent cycle-to-cycle flow variations have received increased attention as a potential stimulus for cardiovascular diseases. These turbulent conditions are typically evaluated in a statistical sense from single scalars extracted from ensemble-averaged tensors (such as the Reynolds stress tensor), limiting the amount of information that can be used for physical interpretations and quality assessments of numerical models. In this study, barycentric anisotropy invariant mapping was used to demonstrate an efficient and comprehensive approach to characterize turbulence-related tensor fields in patient-specific cardiovascular flows, obtained from scale-resolving large eddy simulations. These techniques were also used to analyze some common modeling compromises as well as MRI turbulence measurements through an idealized constriction. The proposed method found explicit sites of elevated turbulence anisotropy, including a broad but time-varying spectrum of characteristics over the flow deceleration phase, which was different for both the steady inflow and Reynolds-averaged Navier–Stokes modeling assumptions. Qualitatively, the MRI results showed overall expected post-stenotic turbulence characteristics, however, also with apparent regions of unrealizable or conceivably physically unrealistic conditions, including the highest turbulence intensity ranges. These findings suggest that more detailed studies of MRI-measured turbulence fields are needed, which hopefully can be assisted by more comprehensive evaluation tools such as the once described herein.

Design of a rear-stage subsonic axial compressor with casing treatments

The aerodynamic design of a 3.5-stage high-pressure axial compressor, representing a modern rear stage design with low aspect ratio blades and large relative tip gaps is presented. The subsonic rotor and stator airfoils feature a controlled-diffusion (CDA) type of sectional design, obtained from 2D viscous-inviscid simulations using the MISES flow solver. Improvements of the main gas path, as well as the 3D blade design, are based on 3D steady Reynolds-averaged Navier–Stokes simulations (ANSYS CFX). Sweep and dihedral were applied to improve the design and off-design performance of the compressor. The matured set of blading, with smooth casing, still exhibits large rotor tip vortices and double leakage, limiting the compressors operability towards stall point. Consequently, the application and development of axial slot casing treatments (CT) into the smooth casing is performed, aimed at improving local stage and overall stall margin with particular emphasis on avoiding efficiency penalties within stages. In order to investigate and optimise the geometric properties of the CTs with respect to efficiency a DoE (Design of Experiments) is executed. Based on the DoE a CT-geometry is deduced and applied to the critical rotor. Due to the fact, that the stage operability with casing treatment is limited by the downstream stator, a first adaption of the existing blade design is considered. In a first step, the downstream stator’s incidence is adjusted, resulting in a slightly increased stall margin. The second step incorporates the modification of rotor blade count, following the idea of trading rotor operability for an efficiency increase, due to reduced friction. However, no notable efficiency benefit was achieved, as the reduced friction loss is counterbalanced by increased secondary flow loss resulting from higher blade loadings.

Effect of undulating blades on highly loaded compressor cascade performance

Humpback whale’s flipper with leading-edge tubercles has been attracting aerodynamic and hydrodynamic researchers’ attentions by its stall-delayed characteristics. Inspired by this, the undulating configuration is used in a highly loaded compressor cascade as a new type of passive flow control technique. A new model of undulating compressor blade is studied in this paper. To investigate the effect of the undulating configuration on cascade performance without the impact of endwall, steady Reynolds-averaged Navier–Stokes simulations of infinite-span cascades are carried out with and without undulations at an inlet Mach number of 0.5. A parametric study is performed to conclude that, with a suitable wavelength, the undulating blade could achieve a rise in diffusion capacity, accompanied by 12.9% reduction in total pressure loss coefficient at a post-stall incidence angle of 8°, whereas it produces negligible impact in cascade performance at 0° incidence angle. Flow visualization further reveals that wavelength is a crucial parameter, determining the spanwise space for the formation of streamwise vortices. Undulating blades could produce positive effects with maximum magnitude when the counter-rotating streamwise vortices take dominant position along span with an appropriate size.

A Variable-Separation Method for Nonlinear Partial Differential Equations With Random Inputs

In this paper, we consider a variable-separation (VS) method to solve the nonlinear partial differential equations (PDEs) with random inputs. The aim of the VS method is to get a sep- arated representation of the Galerkin solution for nonlinear PDEs with random inputs. An essential ingredient of the proposed method is the construction of the optimal stochastic basis functions. The nonlinearity can affect the computation efficiency and may bring challenges for the construction of the optimal stochastic basis functions. In order to overcome the difficulty, we develop the VS method such that the optimal stochastic basis functions are generated in an incremental constructive man- ner. At each enrichment step, a stochastic basis function is determined by the linearized equation deduced from the nonlinear problems at hand. The computation of the VS method decomposes into an offline phase and an online phase. The linearization of the construction for stochastic basis functions can significantly improve the computation efficiency in both offline and online stages. We first describe the VS method for nonlinear stochastic problems in a general framework. Then two nonlinear mod- els with random inputs are considered to formulate the details and methodologies of the proposed method, namely, the nonlinear elliptic equations and the steady Navier--Stokes equations.

DynamFluid: Development and Validation of a New GUI-Based CFD Tool for the Analysis of Incompressible Non-Isothermal Flows

A computational fluid dynamics software (DynamFluid) based on the application of the finite element method with the characteristic-based-split algorithm is presented and validated. The software is used to numerically integrate the steady and unsteady Navier–Stokes equations for both constant-density and Boussinesq non-isothermal flows. Benchmark two-dimensional computations carried out with DynamFluid show good agreement with previous results reported in the literature. Test cases used for validation include (i) the lid-driven cavity flow, (ii) mixed convection flow in a vertical channel with asymmetric wall temperatures, (iii) unsteady incompressible flow past a circular cylinder, and (iv) steady non-isothermal flow past a circular cylinder with negligible buoyancy effects. The new software is equipped with a graphical user interface that facilitates the definition of the fluid properties, the discretization of the physical domain, the definition of the boundary conditions, and the post-processing of the computed velocity, pressure and temperature fields.

Development of High-Fidelity Numerical Methodology Based on Wavenumber-Frequency Transform for Quantifying Internal Aerodynamic Noise in Critical Nozzle

In industrial fields dealing with high-temperature and high-pressure gas such as chemical, petrochemical, and offshore oil production plants, piping systems with valves are frequently used to protect the relevant system and equipment from being damaged by such gases. However, excessive noise is sometimes generated by the valve flow in the piping system, causing so-called acoustic induced vibration in the pipe wall. Therefore, it is of great importance to design the related system to avoid this phenomenon. In this study, a high-fidelity numerical procedure is proposed to assess the acoustic power generated by pressure relief devices in a pipe. The method consists of three sequential steps: high accuracy large eddy simulation, wavenumber-frequency transform, and duct acoustic theory. The critical nozzle is selected as a target system since it is commonly used as a flowmeter and thus there are a lot of relevant data for comparison. First, the steady Reynold-Averaged Navier–Stokes (RANS) solver is used to predict the flow rate of the two-dimensional axisymmetric critical nozzles, and its validity is confirmed by comparing the predicted results with the measured ones. There is good agreement between the two results. Then, a high accuracy Large Eddy Simulation (LES) technique is performed on the three-dimensional critical nozzle, and the steady-state RANS result is used as the initial condition to accelerate the convergence of the unsteady simulation. The validity of the unsteady LES results is also confirmed by comparing them with measured surface pressure data. The wavenumber-frequency transform is taken on the LES results, and the compressible surface pressure components matching the acoustical duct modes are identified in the wavenumber-frequency pressure diagram. The inverse wavenumber-frequency transform taken on the compressible pressure components leads to the acoustic power spectrum. These results reveal that the current numerical procedure can be used to more accurately predict the acoustic power generated by pressure relief device in the piping system.

Analysis of forced convection from a horizontal flat plate for liquid metals

ABSTRACT Numerical simulation of the flow and forced convective heat transfer past a horizontal flat plate for liquid metals has b een performed in elliptical cylindrical coordinates. The two dimensional, steady incompressible Navier–Stokes equations in stream function vorticity formulation together with energy equation are solved using fourth order compact finite difference scheme. A stable and efficient method is introduced to solve the fourth order discretizations of governing partial differential equations through iterations in pseudo time. The fourth order convergence of the present scheme is demonstrated. The drag coefficient, average Nusselt number, variation of skin friction and local Nusselt numbers on the plate surface are shown graphically for Reynolds number up to and Prandtl numbers 0.004, 0.020 and 0.065, respectively for liquid sodium, liquid gallium and liquid lithium. It is found that skin friction coefficient and local Nusselt number approaches to Blasius solution and Pohlhaunsen solution (), respectively as Reynolds number approaches to infinity.

Combined Experimental and CFD Investigation of Flat Plate Film Cooling through Fan Shaped Holes

The present paper reports the results of an experimental and computational investigation of flat plate film cooling jets discharged from three fan-shaped holes. Measurements have been carried out at near unity density ratio in a low-speed wind tunnel, at low inlet turbulence intensity, with blowing ratios (BR) of 1 and 2. Aerodynamic results have shown that the jet stays attached to the flat plate. Thermal measurements have revealed that film cooling effectiveness decreases downstream of the holes, and BR equal to 1 provides the best trade-off between cooling air consumption and thermal protection. Consequently, BR = 1 was selected for assessing the performance of different turbulence models, implemented in STAR-CCM+, according with the steady Reynolds-averaged Navier–Stokes (RANS) approach. Predictions from realizable k-ε (RKE), shear stress transport k-ω (SST KW) and Reynolds stress model (RSM) were compared against measurements of laterally averaged and centerline adiabatic effectiveness, as well as off-the-wall velocity maps and profiles of stress components. RSM provided the most accurate predictions.

Numerical Optimization of a Stall Margin Enhancing Recirculation Channel for an Axial Compressor

A proof of concept is provided by computational fluid dynamic simulations of a new recirculating type casing treatment. This treatment aims at extending the stable operating range of highly loaded axial compressors, so to improve the safety of sorties of high-speed, high-performance aircraft powered by high specific thrust engines. This casing treatment, featuring an axisymmetric recirculation channel, is evaluated on the NASA rotor 37 test case by steady and unsteady Reynolds Averaged Navier Stokes (RANS) simulations, using the realizable k-ε model. Flow blockage at the recirculation channel outlet was mitigated by chamfering the exit of the recirculation channel inner wall. The channel axial location from the rotor blade tip leading edge was optimized parametrically over the range −4.6% to 47.6% of the rotor tip axial chord c z . Locating the channel at 18.2% c z provided the best stall margin gain of approximately 5.5% compared to the untreated rotor. No rotor adiabatic efficiency was lost by the application of this casing treatment. The investigation into the flow structure with the recirculating channel gave a good insight into how the new casing treatment generates this benefit. The combination of stall margin gain at no rotor adiabatic efficiency loss makes this design attractive for applications to high-speed gas turbine engines.

Effect of a Recirculating Type Casing Treatment on a Highly Loaded Axial Compressor Rotor

The tip leakage flow over the blades of an axial compressor rotor adversely affects the axial rotor efficiency and can determine the onset of tip leakage stall. The performance of a new casing treatment concept in the shape of an axisymmetric recirculation channel is explored by steady Reynolds-Averaged Navier–Stokes (RANS) realizable k-ε modelling on the NASA Rotor 37 test case. The modelling exposed a number of attractive features. The casing treatment increased the stall margin at no penalty to the rotor isentropic efficiency over the rotor operating line. A recirculation in the casing channel self-activated and self-adjusted with the rotor loading to provide more passive flow control at higher rotor loading conditions. The nozzle-shaped recirculation channel outflow opposed the tip leakage jet, re-located the casing surface flow interface further downstream, and reduced the rotor blade tip incidence angle. This combination of features makes the new casing treatment particularly attractive for applications to high thrust-to-weight ratio engines, typical of high-performance jet aircraft.

Well-posedness of steady state Navier–Stokes equations with slip boundary conditions

ABSTRACTWe study the motion of a viscous incompressible fluid in a bounded region of containing finitely many rigid solid particles of small size. On the boundary of the particles, we prescribe the slip boundary condition of the form developed in Fujita H. [A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. Mathematical fluid mechanics and modeling (Kyoto, 1994). Srikaisekikenkysho Kkyroku No. 888; 1994. p. 199–216] and Le Roux C, Tani A. [Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions. Math Methods Appl Sci. 2007;30(5):595–624]. We derive a weak formulation of the problem and obtain an equivalent variational inequality formulation. Sufficient conditions for existence, uniqueness, and continuous dependence on data are obtained.

Comparison of Two Methods for Sensitivity Analysis of Compressor Blades

This paper will compare two approaches of sensitivity analysis, namely (i) the adjoint method which is used to obtain an initial estimate of the geometric sensitivity of the gas-washed surfaces to aerodynamic quantities of interest and (ii) a Monte Carlo type simulation with an efficient sampling strategy. For both approaches, the geometry is parameterized using a modified NACA parameterization. First, the sensitivity of those parameters is calculated using the linear (first-order) adjoint model. Since the effort of the adjoint computational fluid dynamics (CFD) solution is comparable to that of the initial flow CFD solution and the sensitivity calculation is simply a postprocessing step, this approach yields fast results. However, it relies on a linear model which may not be adequate to describe the relationship between relevant aerodynamic quantities and actual geometric shape variations for the derived amplitudes of shape variations. Second, in order to better capture nonlinear and interaction effects, a Monte Carlo type simulation with an efficient sampling strategy is used to carry out the sensitivity analysis. The sensitivities are expressed by means of the coefficient of importance (CoI), which is calculated based on modified polynomial regression and therefore able to describe relationships of higher order. The methods are applied to a typical high-pressure compressor (HPC) stage. The impact of a variable rotor geometry is calculated by three-dimensional (3D) CFD simulations using a steady Reynolds-averaged Navier–Stokes model. The geometric variability of the rotor is based on the analysis of a set of 400 blades which have been measured using high-precision 3D optical measurement techniques.

Effect of aspect ratio on the flow and heat transfer characteristics of mist/steam in rectangular ribbed channels

ABSTRACTThe effects of Reynolds number from 10,000 to 80,000, mist mass ratios from 1 to 6%, and droplet sizes from 5 to 20 µm on flow and heat transfer behaviors of mist/steam in rectangular channels with various aspect ratios of 1/4, 1/2, 1/1, 2/1, and the rib angle of 60° are numerically studied in this paper. Additionally, secondary flow distribution in the four ribbed channels and its effect on heat transfer are analyzed in detail. The 3D steady Reynolds-averaged Navier–Stokes equations with a SST k-ω turbulent model are solved by using ANSYS CFX. The CFD model has been verified by the experimental data for steam-only case with a good agreement. The results indicate that similar secondary flow pattern can be observed in the four ribbed channel except for the size of main secondary flow; the heat transfer augmentation of mist/steam raises as Reynolds number and mist mass ratio increase; a peak value of average Nu is obtained in the case of 15 µm mist among all the sizes of droplets. The friction coefficient decays with increase of Reynolds number and mist mass ratio but is insensitive to droplet sizes. The case of AR = 1/2 obtains the best thermal performance in mist/steam cooling channels.

Numerical study of the cross-ventilation of an isolated building with different opening aspect ratios and locations for various wind directions

ABSTRACTIncreasing the natural ventilation potential is an effective way to reduce energy consumption in buildings. In this paper, by using computational fluid dynamics (CFD), the natural ventilation of the wind flow was simulated to evaluate the affecting parameters on the volume flux in an isolated building. The studied parameters were the opening aspect ratios and their locations for various wind directions. Investigations were performed for seven window aspect ratios with the same opening area, six different wind directions, and eight different lateral- and vertical-positions on a leeward wall. The 3D steady Reynolds-averaged Navier–Stokes (RANS) equations were solved by a commercially available finite volume code. To provide closure, an SST k – ω turbulence model was coupled to the momentum equations to evaluate the Reynolds stresses. To validate the numerical method, the CFD results were compared with the numerical and experimental results obtained by other studies. A reasonable agreement was observed between the reference data and this study's results. These results showed that small changes, such as in the window dimensions or the lateral and vertical locations, affected the natural ventilation flow by increasing the volume flux. In addition, these results showed that when the wind direction was at an angle of more than 45°, the volume flow rate was approximately independent of the opening's dimensions.

Simultaneous efficiency improvement of pump and turbine modes for a counter-rotating type pump-turbine

This article presents a multi-objective optimization to improve the hydrodynamic performance of a counter-rotating type pump-turbine operated in pump and turbine modes. The hub and tip blade angles of impellers/runners with four blades, which were extracted through a sensitivity test, were optimized using a hybrid multi-objective genetic algorithm with a surrogate model based on Latin hypercube sampling. Three-dimensional steady incompressible Reynolds-averaged Navier–Stokes equations with the shear stress transport turbulence model were discretized via finite volume approximations and solved on a hexahedral grid to analyze the flow in the pump-turbine domain. For the major hydrodynamic performance parameters, the pump and turbine efficiencies were selected as the objective functions. Global Pareto-optimal solutions were searched using the response surface approximation surrogate model with the non-dominated sorting genetic algorithm, which is a multi-objective genetic algorithm. The trade-off between the two objective functions was determined and described with regard to the Pareto-optimal solutions. As a result, the pump and turbine efficiencies for the arbitrarily selected optimum designs in the Pareto-optimal solutions were increased as compared with the reference design.

Remarks on classical solutions to steady quantum Navier-Stokes equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier–Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier–Stokes equations, when the nonzero boundary value u0 satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u0. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier–Stokes equations.

Global Optimization Under Uncertainty and Uncertainty Quantification Applied to Tractor-Trailer Base Flaps

Using a global optimization evolutionary algorithm (EA), propagating aleatory and epistemic uncertainty within the optimization loop, and using computational fluid dynamics (CFD), this study determines a design for a 3D tractor-trailer base (back-end) drag reduction device that reduces the wind-averaged drag coefficient by 41% at 57 mph (92 km/h). Because it is optimized under uncertainty, this design is relatively insensitive to uncertain wind speed and direction and uncertain deflection angles due to mounting accuracy and static aeroelastic loading. The model includes five design variables with generous constraints, and this study additionally includes the uncertain effects on drag prediction due to truck speed and elevation, steady Reynolds-averaged Navier–Stokes (RANS) approximation, and numerical approximation. This study uses the Design Analysis Kit for Optimization and Terascale Applications (DAKOTA) optimization and uncertainty quantification (UQ) framework to interface the RANS flow solver, grid generator, and optimization algorithm. The computational model is a simplified full-scale tractor-trailer with flow at highway speed. For the optimized design, the estimate of total predictive uncertainty is +15/−42%; 8–10% of this uncertainty comes from model form (computation versus experiment); 3–7% from model input (wind speed and direction, flap angle, and truck speed); and +0.0/−28.5% from numerical approximation (due to the relatively coarse, 6 × 106 cell grid). Relative comparison of designs to the no-flaps baseline should have considerably less uncertainty because numerical error and input variation are nearly eliminated and model form differences are reduced. The total predictive uncertainty is also presented in the form of a probability box, which may be used to decide how to improve the model and reduce uncertainty.