Unsteady Flow Solver Research Articles

A discrete adjoint harmonic balance method for turbomachinery shape optimization

The harmonic balance method has seen an increasing popularity in the solution of time-periodic problems because of its computational efficiency and its ability to model dynamically nonlinear fluid phenomena. In addition, the mathematically steady nature of this technique makes it ideal for adjoint sensitivity analysis of unsteady problems. In this work, a novel optimization framework consisting of three components: a harmonic balance based unsteady cascade flow solver, an accompanying adjoint solver and a quasi-Newton optimization solver; have been developed. The discrete adjoint solver is obtained with the aid of an automatic differentiation tool, TAPENADE. To demonstrate the efficiency and accuracy of the method, we present shape optimization and adjoint sensitivity computations for a two-dimensional compressor cascade. Steady inverse design of this cascade is performed to investigate the effects of two shape parameterization methods, namely Hicks–Henne bump functions and mesh points. Shape optimization is performed to improve the aerodynamic damping characteristics of a vibrating cascade row. In addition, the unsteady adjoint technique is used to determine the frequency of vibration that would drive the system to limit-cycle, which defines the stability limit of the cascade.

Time-varying wing-twist improves aerodynamic efficiency of forward flight in butterflies.

Insect wings can undergo significant chordwise (camber) as well as spanwise (twist) deformation during flapping flight but the effect of these deformations is not well understood. The shape and size of butterfly wings leads to particularly large wing deformations, making them an ideal test case for investigation of these effects. Here we use computational models derived from experiments on free-flying butterflies to understand the effect of time-varying twist and camber on the aerodynamic performance of these insects. High-speed videogrammetry is used to capture the wing kinematics, including deformation, of a Painted Lady butterfly (Vanessa cardui) in untethered, forward flight. These experimental results are then analyzed computationally using a high-fidelity, three-dimensional, unsteady Navier-Stokes flow solver. For comparison to this case, a set of non-deforming, flat-plate wing (FPW) models of wing motion are synthesized and subjected to the same analysis along with a wing model that matches the time-varying wing-twist observed for the butterfly, but has no deformation in camber. The simulations show that the observed butterfly wing (OBW) outperforms all the flat-plate wings in terms of usable force production as well as the ratio of lift to power by at least 29% and 46%, respectively. This increase in efficiency of lift production is at least three-fold greater than reported for other insects. Interestingly, we also find that the twist-only-wing (TOW) model recovers much of the performance of the OBW, demonstrating that wing-twist, and not camber is key to forward flight in these insects. The implications of this on the design of flapping wing micro-aerial vehicles are discussed.

Numerical study of the thunniform mode of fish swimming with different Reynolds number and caudal fin shape

The hydrodynamics of a model-fish swimming in thunniform mode was studied numerically in this paper. A ‘tuna’-like configuration and the undulating manner (the kinetics of swimming) were adopted from some references. The unsteady incompressible RANS equations were solved by an unsteady flow solver based on dynamic hybrid grids, which was developed by the authors in previous work. During the simulations, two typical turbulence models (SA-model and SST-model) were employed to investigate the turbulence effect, and compared with the ‘laminar’ case (switch off the turbulence models). The influence of Reynolds number was studied also. Numerical results demonstrate that the propulsion performance is better when considering turbulence models at higher Reynolds number, because the flow separation is relatively weaker than the ‘laminar’ cases. Furthermore, three types of caudal fin models were considered emphatically, including the popular crescent-shaped fin, a semicircle-shaped fin and a fan-shaped fin. Numerical results show that the crescent-shaped caudal fin is the most efficient when cruising, although the ‘thrust’ is relatively less. The main reason is that the energy loss in the lateral direction is less than those of the other two caudal fin models.

Heat transfer modelling in honeycomb wall-flow diesel particulate filters

Heat transfer in wall-flow monoliths has gained in interest because of the widespread adoption of these systems by automotive industry to fulfil soot emission regulations and the importance of heat exchange on the regeneration process control to avoid damaging the monolith. This paper presents a heat transfer model for wall-flow diesel particulate filters coupled with an unsteady compressible flow solver. The heat exchange between the gas and the solid phase is based on a bi-dimensional discretisation of the porous medium both in axial and tangential directions. The monolith can be discretised in the radial direction to account for the heat fluxes towards the environment through the monolith and the canister, which is also coupled with the inlet and outlet ducts of the filter. The model is validated against experimental data obtained in a flow test rig. A test campaign under non-reacting conditions has been conducted to show the capability for thermal response prediction. Tests cover clean and soot loaded monolith, continuous flow under steady and transient thermal conditions, and pulsating flow. In this case, the characteristics of the pressure waves in amplitude and frequency are similar to those that the monolith can undergo depending on its location along the exhaust line.

Applications of dynamic hybrid grid method for three-dimensional moving/deforming boundary problems

In the previous work, the authors had developed a dynamic hybrid grid generation method and an unsteady flow solver for two-dimensional incompressible and compressible unsteady flows with moving or morphing boundary. In this paper, the dynamic hybrid grid generation method and the unsteady flow solver are extended to three-dimensional complex geometries with moving and/or deforming boundaries, and coupled with force and moment calculation, and the integration of the rigid body, six degrees-of-freedom (6DOF) equations of motion. In order to enhance the flexibility and efficiency of moving grid generation, the dynamic hybrid grid method combines the ‘Delaunay graph’ mapping approach, node relaxation based on ‘spring’ analogy and local re-meshing strategy. Firstly, the prism/tetrahedral/Cartesian hybrid grids are adopted to discrete the initial computational domain over complex configurations. Once the bodies move or deform, the grid points in the boundary layer of the moving/morphing bodies are moved firstly with a modified advancing-layer method, the grid points in the outer far-field keep stationary, while the grid points between the last layer of body-fitted grids and the internal boundary of the specified far-field are mapped by the ‘Delaunay graph’ mapping method. But the background grids (the Delaunay graph) themselves are deformed by the simple node relaxation based on ‘spring’ analogy to improve the efficiency. Then the quality of the deformed grids is checked with some criteria. If the deformed grids do not pass the checking step, a local re-meshing procedure is carried out. Based on the dynamic hybrid grids, a parallel implicit finite-volume flow solver for 3D unsteady Navier–Stokes equations is developed also. In order to deal with the problems of multi-body separation, the integration of the rigid body, 6DOF equations of motion is coupled in the same framework of the flow solver. The applications for complex 3D morphing configurations demonstrate the robustness and efficiency of present method.

AN INTERPOLATION TECHNIQUE FOR THE METHOD OF CHARACTERISTICS IN 4-DIMENSIONS DOMAIN

An interpolation procedure for unsteady inviscid flow (i.e. Euler) solvers is developed. The proposed linear interpolation is first order accurate in time (Δt). The flow properties at each non-mesh point are interpolated in a physical manner as a summation of the values at its surrounding eight mesh points multiplied by their respective weighting coefficients. The coefficients are based on the distances apart between the interpolating non-mesh point and the eight mesh points. The major advantage in this technique is using the position vector for interpolation instead of using the three coordinates x, y & z in succession. This means that it is a one step interpolation instead of three steps. Finally, the compatibility is achieved from using the hyper characteristic curves to locate the initial data points. The proposed technique offers an accurate, efficient and compatible interpolation valid for the method of characteristics in 4-D.

Concurrent Blade Aerodynamic-Aero-elastic Design Optimization Using Adjoint Method

Increasing aerothermal and aero-elastic performance requirements and constraints are closely linked in modern blading designs. There is thus a need for more concurrent interaction between the disciplines at earlier stages of a design process. Presented in this paper are the development, validation, and demonstration of the adjoint approach to concurrent blading aerodynamic and aero-elastic design optimizations. A nonlinear harmonic phase solution method is adopted to solve the unsteady Reynolds-averaged Navier–Stokes equations. The flow field response in terms of both the mean aerothermal performance and aero-elastic stability to a geometrical perturbation can be obtained by three “steadylike” flow solutions at three distinctive temporal phases. This unsteady flow solution method is computationally very efficient and provides a convenient and consistent base for formulating the corresponding adjoint equations. The adjoint system for the unsteady flow solver is solved effectively by a relatively simple extension of the method and techniques previously developed for a steady flow adjoint solver. As a result, the sensitivities of both the steady (time-mean) flow loss and the aerodynamic damping/forcing to detailed blade geometry changes can be very efficiently obtained by solving equivalently three steadylike adjoint equations. Several case studies are presented to illustrate the validity and effectiveness of this new concurrent approach.

Simulation of rotating stall in a whole stage of an axial compressor

Aerodynamic instabilities that naturally occur in compression systems, such as surge and rotating stall, largely reduce the life duration and performance of system components. The prediction of the compressor operating range is thus a key parameter for the design of gas turbines. This paper investigates the ability of an unsteady flow solver to simulate the rotating stall phenomenon in the full annulus of an axial compressor stage. A comparison with experimental data indicates that the simulation correctly estimates the stability limit. However the rotating stall flow patterns are different. While measurements show only one full span rotating stall cell (40 Hz), the simulation shows first a part span stall with 10 cells (790 Hz) that evolves then towards a full span stall with three cells (170 Hz). A spectral analysis based on numerical results underlines the role of rotor–stator interactions in the development of rotating stall. The effects of downstream volumes and inlet distortions are also discussed, showing the necessity to consider the whole geometry to correctly predict the rotating stall frequency.

Experimental investigation of the effect of chordwise flexibility on the aerodynamics of flapping wings in hovering flight

Ornithopters or mechanical birds produce aerodynamic lift and thrust through the flapping motion of their wings. Here, we use an experimental apparatus to investigate the effects of a wing's twisting stiffness on the generated thrust force and the power required at different flapping frequencies. A flapping wing system and an experimental set-up were designed to measure the unsteady aerodynamic and inertial forces, power usage and angular speed of the flapping wing motion. A data acquisition system was set-up to record important data with the appropriate sampling frequency. The aerodynamic performance of the vehicle under hovering (i.e., no wind) conditions was investigated. The lift and thrust that were produced were measured for different flapping frequencies and for various wings with different chordwise flexibilities. The results show the manner in which the elastic deformation and inertial flapping forces affect the dynamical behavior of the wing. It is shown that the generalization of the actuator disk theory is, at most, only valid for rigid wings, and for flexible wings, the power P varies by a power of about 1.0 of the thrust T. This aerodynamic information can also be used as benchmark data for unsteady flow solvers.

Harmonic Balance Analysis of Blade Row Interactions in a Transonic Compressor

In this paper we apply the harmonic balance technique to analyze an inlet guide vane and rotor interaction problem, and compare the computed flow solutions to existing experimental data. The computed results, which compare well with the experimental data, demonstrate that the technique can accurately and efficiently model strongly nonlinear periodic flows, including shock/vane interaction and unsteady shock motion. Using the harmonic balance approach, each blade row is modeled using a computational grid spanning just a single blade passage regardless of the actual blade counts. For each blade row, several subtime level solutions that span a single time period are stored. These subtime level solutions are related to each other through the time derivative term in the Euler (or Navier―Stokes) equations, which is approximated by a pseudo-spectral operator, by complex periodicity conditions along the periodic boundary of each blade row's computational domain, and by the interface boundary conditions between the vane and rotor. Casting the governing equations in harmonic balance form removes the explicit dependence on time. Mathematically, the equations to be solved are similar in form to the steady Euler (or Navier―Stokes) equations with an additional source term proportional to the fundamental frequency of the unsteadiness. Thus, conventional steady-state computational fluid dynamics techniques, including local time stepping and multigrid acceleration, are used to accelerate convergence, resulting in a very efficient unsteady flow solver.

Experimental study on interaction of aerodynamics with flexible wings of flapping vehicles in hovering and cruise flight

Flapping wings are promising lift and thrust generators, especially for very low Reynolds numbers. To investigate aeroelastic effects of flexible wings (specifically, wing’s twisting stiffness) on hovering and cruising aerodynamic performance, a flapping-wing system and an experimental setup were designed and built. This system measures the unsteady aerodynamic and inertial forces, power usage, and angular speed of the flapping wing motion for different flapping frequencies and for various wings with different chordwise flexibility. Aerodynamic performance of the vehicle for both no wind (hovering) and cruise condition was investigated. Results show how elastic deformations caused by interaction of inertial and aerodynamic forces with the flexible structure may affect specific power consumption. This information was used here to find a more suitable structural design. The best selected design in our tests performs up to 30% better than others (i.e., less energy consumption for the same lift or thrust generation). This measured aerodynamic information could also be used as a benchmarking data for unsteady flow solvers.

Unsteady Simulations of Bell-Agusta 609 Rotor Undergoing Higher Harmonic Oscillation

The unsteady flowfield of a Bell-Agusta 609 rotor is numerically investigated with an unsteady Reynolds-averaged Navier-Stokes flow solver based on unstructured grid topology. The geometry considered is a three-bladed rotor that is undergoing a higher harmonic blade-pitch oscillation. Methodology is presented to model the multiblade cyclic motion that is superimposed on the basic motion. The unsteady airloads and blade-vortex interaction phenomenon are investigated for hovering and forward flight conditions. Numerical simulations indicate a significant impact of the blade-pitch oscillation on the tip-vortex structure and associated blade-vortex interaction.

On the Influence of Far-Wake Resolution on Wind Turbine Flow Simulations

A study was made on the influence of the wake mesh resolution on the integrated forces acting on a wind turbine rotor using an unsteady incompressible Navier-Stokes flow solver. Solutions were obtained for five wake resolutions ranging from 0.5 rotor diameters to 7.5 diameters downstream resolving up to 13 tip vortex revolutions. It was found that the resolution of the tip vortices in the far-wake beyond half a diameter downstream did not influence the induced velocities in the rotor plane significantly, with the low speed shaft torque differing less than approximately 1% for the rotor in question.

단분리 시스템의 분리 거동 해석

Separation motion analysis of staging system is conducted using combined analysis programs, which include unsteady aerodynamic analysis codes and dynamic motion analysis tools. In this study, the analysis is for the long-rang missile staging system. The purpose of this study is to verify the safety and reliance of the proposed staging system, and to find out the influence of angle of attack perturbation on staging. A structured parallel overset mesh called Chimera grid is used for the simulation of unsteady supersonic Euler flow solver. In addition, unsteady dynamic simulations are also performed.

A Numerical Method for Turbomachinery Aeroelasticity

This work provides an accurate and efficient numerical method for turbomachinery flutter. The unsteady Euler or Reynolds-averaged Navier-Stokes equations are solved in integral form, the blade passages being discretised using a background fixed C-grid and a body-fitted C-grid moving with the blade. In the overlapping region data are exchanged between the two grids at every time step, using bilinear interpolation. The method employs Roe’s second-order-accurate flux difference splitting scheme for the inviscid fluxes, a standard second-order discretisation of the viscous terms, and a three-level backward difference formula for the time derivatives. The dual-time-stepping technique is used to evaluate the nonlinear residual at each time step. The state-of-the-art second-order accuracy of unsteady transonic flow solvers is thus carried over to flutter computations. The code is proven to be accurate and efficient by computing the 4th Aeroelastic Standard Configuration, namely, the subsonic flow through a turbine cascade with flutter instability in the first bending mode, where viscous effect are found practically negligible. Then, for the very severe 11th Aeroelastic Standard Configuration, namely, transonic flow through a turbine cascade at off-design conditions, benchmark solutions are provided for various values of the inter-blade phase angle.

A solver using Newton's method for unsteady viscous flows

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton's method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, i.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart-Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, i.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of time-derivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.

Predicting Bladerow Interactions Using a Multistage Time-Linearized Navier-Stokes Solver

A multistage frequency domain (time-linearized/nonlinear harmonic) Navier-Stokes unsteady flow solver has been developed for predicting unsteady flows induced by bladerow interactions. In this paper, the time-linearized option of the solver has been used to analyze unsteady flows in a subsonic turbine test stage and the DLR transonic counter-rotating shrouded propfan. The numerical accuracy and computational efficiency of the time-linearized viscous methods have been demonstrated by comparing predictions with test data and nonlinear time-marching solutions for these two test cases. It is concluded that the development of efficient frequency domain approaches enables unsteady flow predictions to be used in the design cycles to tackle aeromechanics problems.

High-resolution viscous flow simulations at arbitrary Mach number

A characteristic-based unsteady viscous flow solver is developed with preconditioning that is uniformly applicable for Mach numbers ranging from essentially incompressible to supersonic. A preconditioned flux-difference formulation for nondimensional primitive variables is a key element of the present approach. The simple primitive-variable numerical flux is related to Roe’s flux-difference scheme and preserves contact discontinuities using primitive variables, with or without preconditioning. Preconditioning by a single-parameter diagonal matrix conditions the system eigenvalues in terms of nondimensional local velocity and local temperature. An iterative implicit solution algorithm is given for the preconditioned formulation and is used for several simple test and validation cases. These include an inviscid shock-tube case, flat-plate boundary layer flow at low Mach number, viscous flow past a circular cylinder at low Reynolds number and with different thermal boundary conditions, and validation cases for incompressible and transonic flows.

An algebraic grid motion technique for large deformations

Numerical simulation of the aeroelastic behaviour of wings and rotor blades requires the coupling of an unsteady flow-solver with a structural model. The moving and deforming surfaces resulting from unsteady simulations require deforming meshes during the simulation, and it is common practice to use simple interpolation of surface displacements and velocities onto the initial undisturbed mesh. However, such techniques result in poor grid quality, and possible grid crossover, when applied to large surface displacements or deformations. A new interpolation technique, for structured grids, is presented which is still simple in that it is driven solely by surface motion, but maintains grid quality near the surface by representing rotational effects there. Furthermore, since the scheme is fully analytic, the grid speeds required in an unsteady flow-solver are also available analytically, and the scheme is very cheap computationally.

A linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems in all flow regimes

We present a computational fluid dynamics (CFD)-based linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems. This method is valid in the subsonic, transonic, and supersonic flow regimes, and is insensitive to the frequency or damping level of the sought-after coupled eigenmodes. It is based on the solution by an orthogonal iteration procedure of a complex eigenvalue problem derived from the linearization of a three-field fluid/structure/moving mesh formulation. The key computational features of the proposed method include the reuse of existing unsteady flow solvers, a second-order approximation of the flux Jacobian matrix, and a parallel domain decomposition-based iterative solver for the solution of large-scale systems of discretized fluid/structure equations. While the frequency analysis method proposed here is primarily targeted at the extraction of the eigenpairs of a wet structure, we validate it with the flutter analysis of the AGARD Wing 445.6, for which experimental data is available.