Discrete Adjoint Solution of Unsteady Turbulent Flow in Compressor

This paper presents an adjoint method for the optimal control of unsteady flows. The goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time-accurate method. First, this paper presents the complete formulation of the time-dependent optimal design problem. Second, we present the time-accurate unsteady continuous and discrete adjoint equations. Third, we present results that demonstrate the application of the theory to a two-dimensional oscillating airfoil. The results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization.

The unsteady convective flow effects in a transonic compressor rotor with a circumferential-groove casing treatment are investigated in this paper. Experimental results show that the circumferential-groove casing treatment increases the compressor stall margin by almost 50% for the current transonic compressor rotor. Steady flow simulation of the current casing treatment, however, yields only a 15% gain in stall margin. The flow field at near-stall operation is highly unsteady due to several self-induced flow phenomena. These include shock oscillation, vortex shedding at the trailing edge, and interaction between the passage shock and the tip clearance vortex. The primary focus of the current investigation is to assess the effects of flow unsteadiness and unsteady flow convection on the circumferential-groove casing treatment. Unsteady Reynolds-averaged Navier-Stokes (URANS) and Large Eddy Simulation (LES) techniques were applied in addition to steady Reynolds-averaged Navier-Stokes (RANS) to simulate the flow field at near-stall operation and to determine changes in stall margin. The current investigation reveals that unsteady flow effects are as important as steady flow effects on the performance of the circumferential grooves casing treatment in extending the stall margin of the current transonic compressor rotor. The primary unsteady flow mechanism is unsteady flow injection from the grooves into the main flow near the casing. Flows moving into and out of the grooves are caused due to local pressure difference near the grooves. As the pressure field becomes transient due to self-induced flow oscillation, flow injection from the grooves also becomes unsteady. The unsteady flow simulation shows that this unsteady flow injection from the grooves is substantial and contributes significantly to extending the compressor stall margin. Unsteady flows into and out of the grooves have as large a role as steady flows in the circumferential grooves. While the circumferential-groove casing treatment seems to be a steady flow device, unsteady flow effects should be included to accurately assess its performance as the flow is transient at near-stall operation.

Three-dimensional dynamic gas–liquid flow simulations that accurately track the phase interface are numerically challenging. This article presents a numerical study of the performance of the level-set phase–interface tracking method when combined with extremely high order (7th to 11th) weighted essentially non-oscillatory (WENO) advection schemes for gas–liquid free surface flows. Comparisons between simulation results and prior benchmark results suggest that such a combination of methods can be satisfactorily applied to the level-set and Navier-Stokes equations for free surface flow simulations when volume conservation is enforced at every time step, and minor numerical oscillations are suppressed through use of an artificial viscosity term. In particular, simulations of solid body rotation, the unsteady flow following an ideal dam break, tank sloshing, and the rise of a single bubble all agree with analytical or experimental results to within ± 3.12% when the level-set method is combined with an 11th order WENO scheme. Furthermore, use of an 11th order WENO advection scheme actually has a computational cost advantage because, for the same accuracy, it can be used on a coarser grid when compared with a more-common second-order advection scheme; computational savings of up to 87% are possible.

Unsteady turbulent flow simulations on moving Cartesian grids using immersed boundary method and high-order scheme

Indices of the intra-aneurysm hemodynamic environment have been proposed as potentially indicative of their longitudinal outcome. To be useful, the indices need to be used to stratify large study populations and tested against known outcomes. The first objective was to compile the diverse hemodynamic indices reported in the literature. Furthermore, as morphology is often the only patient-specific information available in large population studies, the second objective was to assess how the ranking of aneurysms in a population is affected by the use of steady flow simulation as an approximation to pulsatile flow simulation, even though the former is clearly non-physiological. Sixteen indices of aneurysmal hemodynamics reported in the literature were compiled and refined where needed. It was noted that, in the literature, these global indices of flow were always time-averaged over the cardiac cycle. Steady and pulsatile flow simulations were performed on a population of 198 patient-specific and 30 idealised aneurysm models. All proposed hemodynamic indices were estimated and compared between the two simulations. It was found that steady and pulsatile flow simulations had a strong linear dependence (r ≥ 0.99 for 14 indices; r ≥ 0.97 for 2 others) and rank the aneurysms in an almost identical fashion (ρ ≥ 0.99 for 14 indices; ρ ≥ 0.96 for other 2). When geometry is the only measured piece of information available, stratification of aneurysms based on hemodynamic indices reduces to being a physically grounded substitute for stratification of aneurysms based on morphology. Under such circumstances, steady flow simulations may be just as effective as pulsatile flow simulation for estimating most key indices currently reported in the literature.

Development and assessment of adjoint sensitivity analysis method for transient two-phase flow simulations

PurposeTo perform 3D unsteady Reynolds Averaged Navier‐Stokes (URANS) simulations to predict turbulent flow over bluff body.Design/methodology/approachTurbulence closure is achieved through a non‐linear k−ε model. This model is incorporated in commercial FLUENT software, through user defined functions (UDF).FindingsThe study shows that the present URANS with standard wall functions predicts all the major unsteady phenomena, with a good improvement over other URANS reported so far, which incorporate linear eddy viscosity models. The results are also comparable with those obtained by LES for the same test case.Originality/valueWhen comparing the computational time required by the present model and by LES, the accuracy achieved is significant and can be used for simulating 3D unsteady complex engineering flows with reasonable success.

We present a novel sensitivity analysis for predicting the effect of airfoil shape on the unsteady aerodynamic and aeroacoustic response of turbomachinery blading. The nominal steady and unsteady e ow in a cascade of turbomachinery blades is modeled using the steady Euler equations and the time-linearized Euler equations, respectively.BoththesteadyandunsteadyEulerequationsaresolvedusing atwo-stepe nite-volumeLax ‐Wendroff discretizationwithmultigrid acceleration.Wecomputetheunsteady aerodynamicloadsduetoboth incominggusts and plunging motion of the cascade airfoils. Once the nominal steady and unsteady e ows have been computed, a sensitivity analysis is performed using the discrete adjoint equations of the computational e uid dynamics scheme used to discretizetheEulerequations. For each objectivefunction (e.g., the amplitude peak of theaeroelastic blade motion),theresultingadjointequationsaresolvedusingtheadjointLax ‐Wendroffscheme,whichisalsoaccelerated using a multigrid technique. Once the adjoint equations have been solved, the computed adjoint variables may be used to compute rapidly the sensitivities of the aeroelastic and aeroacoustic objective functions due to arbitrary changes in geometry. The method is computationally efe cient, with similar convergence rate histories for both the nominal and the adjoint solutions. To demonstrate the utility of the present method, we use the sensitivity analysis to redesign the shape of the airfoils of a cascade for increased aeroelastic stability. We also redesign the shape of the airfoils of an exit guide vane for reduced downstream radiated noise.

A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations

The performance of radial diffusers in centrifugal compressor stages is influenced by the impeller exit flow characteristics as well as the vaneless and semi-vaneless space flow characteristics. Both steady and unsteady flow interactions are present due to the propagation of pressure fields upstream and downstream between the impeller and diffuser. Furthermore, unsteady flow interactions occur when the impeller moving wakes developed due to secondary and tip clearance flows propagate through the diffuser passages. The present study aims at presenting a model that describes the unsteady wake propagation in the centrifugal compressor diffuser using vorticity principles. 3D unsteady Reynolds-Averaged Navier Stokes simulations are performed for both a vaned and a vaneless diffuser centrifugal stage. The simulations are used to examine the mechanism for the unsteady wake flow interactions in the diffuser. The unsteady streamwise vortical structures present in the impeller wakes and their propagation through the diffuser flow field is presented. The effect of the unsteady flow on loss production in the compressor stage is investigated. The velocity perturbations due to these vortical structures are presented. The present study indicates that the impeller wake propagation in the diffuser can be represented by a series of unsteady streamwise vortices superimposed on a uniform flow field. These vortices result in velocity perturbations that take the form of both positive and negative jets in the diffuser.

A two-dimensional numerical model is developed to predict the time variation of bed deformation in alluvial channel bends. In this model, the depth-averaged unsteady water flow equations along with the sediment continuity equation are solved by using the Beam and Warming alternating-direction implicit scheme. Unlike the present models based on Cartesian or cylindrical coordinate systems and steady flow equations, a body-fitted coordinate system and unsteady flow equations are used so that unsteady effects and natural channels may be modeled accurately. The effective stresses associated with the flow equations are modeled by using a constant eddy-viscosity approach. This study is restricted to beds of uniform particles, i.e., armoring and grain-sorting effects are neglected. To verify the model, the computed results are compared with the data measured in 140° and 180° curved laboratory flumes with straight reaches up- and downstream of the bend. The model predictions agree better with the measured data tha...

The unsteady behaviour of delta wing vortical flows is still a subject which is a challenge for numerical methods, such as computational fluid dynamics. New approaches to turbulence modelling, such as detached eddy simulation (DES) have been proposed which allow for greater accuracy of the numerical predictions. However, this increase in accuracy comes with a considerable increase in computational expense compared to traditional turbulence modelling. This investigation considers the use of both DES and unsteady Reynolds averaged Navier-Stokes (URANS) turbulence methods in the prediction of unsteady vortical flows. Calculations using DES are initially considered for two test cases and the predicted behaviour and resolution of the flow are considered. These results are then validated against existing experimental data, before the URANS calculations are considered with respect to the DES results. From this investigation it is determined that URANS turbulence models are able to predict the dominant features of the low frequency phenomenon present in the vortex system and in the post-breakdown region. Consideration is also given to the effect of grid refinement for both methods and the effect of temporal resolution for the DES calculations.

This paper presents an analysis of refinement indicators for the simulation of steady and unsteady flows by means of p-adaptive algorithms for discontinuous Galerkin (DG) methods. Residual-error, discretization-error and feature-based indicators are compared by studying their effect on convergence history, computational gain and refinement regions selected by the adaptive algorithm. The analysis is initially carried out on steady flow configurations. Static p-adaptive simulations of the periodic flow past a cylinder at Re=100 are then performed. Compared to uniform p-refinement, a reduction between 50% and 75% in the number of degrees of freedom is obtained for all test cases considered. The accuracy and efficiency observed for the small-scale energy density [1] and spectral decay [2] indicators demonstrate their great potential for the adaptive simulation of unsteady flows.

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows. General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived. With the adjoint method, the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function, regardless of the number of design parameters. An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method. Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil, wing, and wing-body configuration, and the aerodynamic performance improvement of turbine and compressor blade rows. The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.

Rate optimization of fractional flow reservoir model based on the continuous adjoint method