Prescribed Pressure Distribution Research Articles

Hybrid inverse/optimization design method for rigid coaxial rotor airfoils considering reverse flow

Double-ended airfoils with round leading and trailing edges outperform conventional round-nose and sharp trailing-edge airfoils when used for the inboard part of a rigid coaxial rotor operating in forward flight, as reverse flow can occupy 80% surface of the retreating blade. However, aerodynamic design optimization for this kind of airfoils is challenging, since it is required to achieve drag reduction over a wider lift range in both forward and reverse flows but there is no method available currently. To address this problem, this article proposes an efficient hybrid inverse/optimization design method that combines inverse design and direct optimization within a surrogate-based optimization (SBO) framework. The weighted objective function consists of two components: prescribed pressure distributions and total drag. The resulting constrained optimization problem is solved by using kriging surrogate model and dedicated infill-sampling method. The proposed method is demonstrated by aerodynamic shape optimizations of a DBLN-526 airfoil and an elliptical airfoil and compared with direct optimization method. For DBLN-526 airfoil case, 7%–30% drag reduction is achieved in the range from zero lift to maximum lift and the optimum airfoil features better stall characteristics. For the case of elliptical airfoil of 40% thickness, significant drag reduction is achieved in both forward and reverse flows and the optimum airfoil features more robust aerodynamic performance at off-design Mach numbers. The results confirm that the proposed method outperforms the conventional direct optimization method and more practical for aerodynamic shape optimization of double-ended airfoils.

A pressure-controllable bump based on the pressure-ridge concept

An innovative type of pressure distribution for the hypersonic aircraft forebody (bump) is presented, and this design is based on the newly established known as pressure ridge (PR) flow mechanism. Studies on the low-kinetic-energy fluid over the aircraft surface are summarized. Then, the challenges of the inlet–airframe integration at high speeds are discussed. The flow structure around the bump is analyzed in detail to eliminate the side-embedded shock (SES) effect at the inlet entrance. The concept of PR is proposed to improve the overall aerodynamic characteristics of the bump, namely, the boundary layer removal, the reduction of external drag, and the streamline direction at bump end-section. On the basis of the developed inverse method to generate the bump surface by the prescribed pressure distribution, the improved PR-derived bump is designed and numerically compared with the typical pressure-controllable bump (PCB) while the identical leading edge profile is imposed. Results demonstrate that the PR creates outward and inward pressure gradients. The removed amount of boundary layer increases with the increase in outward pressure gradient. Meanwhile, the inward pressure gradient determines the streamline pattern after the bump. The PR-derived bump is 29.2% lower in height than the typical one by using a proper outward pressure gradient. The uniform area of the new bump is 30% wider than that of the typical PCB. The near-wall streamlines of the new bump are adjusted from expanding to the parallel, thereby relieving the side-compression. Changing the location, width, and peak value of the PR can lead to great flexibility in the design and optimization of aircraft forebody subjected to hypersonic flow.

Heuristic pressure profiles in hydraulic fracturing

AbstractIn hydraulic fracturing, the pressure exerted by the fracking fluid onto the surrounding solid, is typically obtained from solving the Reynolds equation. This is not always robust and leads to a complex behaviour at the crack front (toughness or viscous regimes). In the presented work, the Reynolds equation is replaced by a simplified fluid model, where pre‐defined pressure distributions are assumed which lead to a simpler and robust coupled problem. The mapping of dimensionless prescribed pressure distributions onto the physical crack surface of the mesh is discussed based on a dimensionless distance function. This function is obtained by solving the Laplace‐Beltrami operator on the explicit representation of the manifold, i.e., the surface mesh, where distances are evaluated along streamlines which represent the fluid flow of the fracking fluid within the fracture.

Inverse design and Mach 6 experimental investigation of a pressure controllable bump

The low kinetic flow starting from the airframe leading edge is a neglected aspect in designing hypersonic air-breathing flight vehicles. Compared with other boundary layer treating technologies, the bump concept can obtain a good balance of boundary layer removal, external drag control, shock system simplification, and integration design flexibility capabilities. On the basis of conventional conical-flow theory and the new 3D inverse design method, this study proposes a pressure controllable bump concept that can generate the bump configuration inversely by the prescribed pressure distribution. The bump/inlet integration pattern is analyzed, and the basic design methodology is presented. To validate whether the pressure distribution can be used in diverting the boundary layer, experimental study of the bump and numerical simulation are conducted. Results show that the bump has generated identical pressure distribution to the design. The bump can also divert approximately 50% of the boundary layer from the incoming low kinetic flow in Mach 6. Compared with the conventional cone-derived bump in Mach 6, the new bump is 25.8% shorter in height. The flow structure is adjusted nearly parallel to the x-direction, thereby promoting the flow quality of the inlet entrance. Hence, the new inverse design method of pressure distribution expands the applicable Mach range of the hypersonic airframe forebody.

METHOD OF SOLVING THE INVERSE PROBLEM FOR WING–FUSELAGE COMBINATION USING REYNOLDS-AVERAGED NAVIER–STOKES EQUATIONS

In this study, an iteration method has been developed for construction of the wing geometry using a prescribed pressure distribution for a wing–fuselage combination in the compressible flow of a viscous gas within the framework of the Reynolds-averaged Navier–Stokes equations. The method belongs to the class of residual corrections methods. The wing surface deformation reducing the residual correction between the computed and objective pressure distribution is determined by solving the inverse problem for the same combination in the potential flow of a compressible gas by taking into account the effect of the boundary layer displacement thickness and the wake behind the wing. Examples of the wing geometry construction using the prescribed pressure distribution and demonstrating the high rate of method convergence (3–4 iterations) are presented.

Subsonic and transonic airfoil inverse design via Ball-Spine Algorithm

Inverse design in external flow regimes usually involves finding the wall shape associated with a prescribed distribution of wall pressure or velocity. In this research, a novel iterative inverse design method is developed for inviscid subsonic and transonic external flow regimes. The method links up a novel inverse design algorithm, called Ball-Spine Algorithm (BSA), and a 2D inviscid analysis code. The Euler equations are solved for a physical domain of which some unknown boundaries are iteratively modified via BSA until a prescribed pressure distribution is reached. In BSA, the unknown walls are composed of a set of virtual balls that move freely along the specified directions called spines. The difference between target and current pressure distributions causes to deform the flexible boundary at each modification step. Some validation test cases and design examples in subsonic and transonic regimes are presented here which show the robustness and flexibility of the method in handling the complex geometries in various flow regimes.

Centrifugal compressor shape modification using a proposed inverse design method

This paper is concerned with a quasi-3D design method for the radial and axial diffusers of a centrifugal compressor on the meridional plane. The method integrates a novel inverse design algorithm, called ball-spine algorithm (BSA), and a quasi-3D analysis code. The Euler equation is solved on the meridional plane for a numerical domain, of which unknown boundaries (hub and shroud) are iteratively modified under the BSA until a prescribed pressure distribution is reached. In BSA, unknown walls are composed of a set of virtual balls that move freely along specified directions called spines. The difference between target and current pressure distributions causes the flexible boundary to deform at each modification step. In validating the quasi-3D analysis code, a full 3D Navier-Stokes code is used to analyze the existing and designed compressors numerically. Comparison of the quasi-3D analysis results with full 3D analysis results shows viable agreement. The 3D numerical analysis of the current compressor shows a huge total pressure loss on the 90° bend between the radial and axial diffusers. Geometric modification of the meridional plane causes the efficiency to improve by about 10%.

Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems

AbstractIn this paper, four approaches to compute the Hessian matrix of an objective function used often in aerodynamic inverse design problems are presented. The computationally less expensive among them is selected and applied to the reconstruction of cascade airfoils that reproduce a prescribed pressure distribution over their walls, under inviscid and viscous flow considerations. The selected approach is based on the direct sensitivity analysis method for the computation of first derivatives, followed by the discrete adjoint method for the computation of the Hessian matrix. The applications presented in this paper show that the Newton method, based on exact Hessian matrices, outperforms other gradient‐based algorithms such as steepest descent or BFGS algorithm. Copyright © 2007 John Wiley & Sons, Ltd.

Design of ship's bow form by potential-based panel method

A surface panel method treating a boundary-value problem of the Dirichlet type with the free surface is presented to design a three-dimensional body corresponding to a prescribed pressure distribution. The free surface boundary condition is linearized with respect to the oncoming flow, and computed by four-point finite difference scheme. Sample designs for submerged spheroids and Wigley hull are carried out to demonstrate the stable convergence, the effectiveness and the robustness of the method. The design of a 5500TEU container carrier is performed with respect to reduction of the wave resistance. To reduce the wave resistance, calculated pressure on the hull surface is modified to have the lower fluctuation, and is applied as a Dirichlet type dynamic boundary condition on the hull surface. The designed hull form is verified to have the lower wave resistance than the initial one not only by computation but also by experiment.

Design and optimization method for a two-dimensional hydrofoil

Abstract In this paper, methods for the design and the optimization of two-dimensional hydrofoils are presented. First a previously developed potential flow design method is extended to the viscous flow computations. In this design method, the foil geometry is defined by B-splines, and the RANS method is utilized for the viscous flow computations. The foil is designed based on a prescribed pressure distribution, and the Newton-Raphson method is used to achieve a foil geometry generating the prescribed pressure distribution. In the design method described above, although designers can specify a pressure distribution based on the desired local characteristics, it is hard to know the forces on the desired foil in advance. Therefore, an “optimization method” by using the Lagrange multiplier method is then developed. The optimization problem described in the proposed paper is to satisfy the lift requirement by minimizing the drag force. The design variables are the angle of attack and the “geometry parameters”. In the presented design examples, the camber distribution and the pressure distribution “shape function” are selected as the geometry parameter. Both the “design” and the “optimi- zation” methods can be used as a design tool individually; however, the combination of two methods provides a useful design tool that can not only optimize the global performance, but also design the detailed geometry according to designers' requirements. Design examples are illustrated in the paper, and this design method is proved to be practical and effective.

Inverse design of supersonic diffuser with flexible walls using a Genetic Algorithm

An efficient algorithm for the design optimization of the compressible fluid flow problem through a flexible structure is presented. The methodology has three essential parts: first the behavior of compressible flow in a supersonic diffuser was studied numerically in quasi-one-dimensional form using a flux splitting method. Second, a fully coupled sequential iterative procedure was used to solve the steady state aeroelastic problem of a flexible wall diffuser. Finally, a robust Genetic Algorithm was implemented and used to calculate the optimum shape of the flexible wall diffuser for a prescribed pressure distribution.

Direct Design of Ducts

This paper describes a method for calculating the shape of duct that leads to a prescribed pressure distribution on the duct walls. The proposed design method is computationally inexpensive, robust, and a simple extension of existing computational fluid dynamics methods; it permits the duct shape to be directly calculated by including the coordinates that define the shape of the duct wall as dependent variables in the formulation. This “direct design method” is presented by application to two-dimensional ideal flow in ducts. The same method applies to many problems in thermofluids, including the design of boundary shapes for three-dimensional internal and external viscous flows.

Metal-forming process optimisation by inverse evolutionary search

In this paper, new numerical techniques are developed for the optimisation of steady and non-steady forming processes. These techniques consider the inverse problem formulation together with evolutionary search schemes. The solutions are obtained using numerical algorithms based on a genetic search supported by an elitist strategy. In order to demonstrate the efficiency of the inverse evolutionary search, two specific cases are considered. The first case involves the optimisation of the process parameters aiming for a prescribed pressure distribution at the exit of the extrusion chamber. In the second case, the optimisation objective aims at the reduction of the difference between the realised and the prescribed final forged shape under minimal energy consumption.

Flow structure and modeling issues in the closure region of attached cavitation

Particle image velocimetry (PIV) and high-speed photography are used to measure the flow structure at the closure region and downstream of sheet cavitation. The experiments are performed in a water tunnel of cross section 6.35×5.08 cm2 whose test area contains transparent nozzles with a prescribed pressure distribution. This study presents data on instantaneous and averaged velocity, vorticity and turbulence when the ambient pressure is reduced slightly below the cavitation inception level. The results demonstrate that the collapse of the vapor cavities in the closure region is the primary mechanism of vorticity production. When the cavity is thin there is no reverse flow downstream and below the cavitation, i.e., a reentrant flow does not occur. Instead, the cavities collapse as the vapor condenses, creating in the process hairpin-like vortices with microscopic bubbles in their cores. These hairpin vortices, some of which have sizes as much as three times the height of the stable cavity, dominate the flow downstream of the cavitating region. The averaged velocity distributions show that the unsteady collapse of the cavities in the closure region involves substantial increase in turbulence, momentum, and displacement thickness. Two series of tests performed at the same velocity and pressure, i.e., at the same hydrodynamic conditions, but at different water temperatures, 35 °C and 45 °C, show the effect of small changes in the cavitation index (σ=4.69 vs. σ=4.41). This small decrease causes only a slight increase in the size of the cavity, but has a significant impact on the turbulence level and momentum deficit in the boundary layer downstream. Ensemble averaging of the measured instantaneous velocity distributions is used for estimating the liquid void fraction, average velocities, Reynolds stresses, turbulent kinetic energy and pressure distributions. The results are used to examine the mass and momentum balance downstream of the cavitating region. It is shown that in dealing with the ensemble-averaged flow in the closure region of attached cavitation, one should account for the sharp (but still finite) gradients in the liquid void fraction. The 2-D continuity equation can only be satisfied when the gradients in void fraction are included in the analysis. Using the momentum equation it is possible to estimate the magnitude of the “interaction term,” i.e., the impact of the vapor phase on the liquid momentum. It is demonstrated that, at least for the present test conditions, the interaction term can be estimated as the local pressure multiplied by the gradient in void fraction.

A new compressor and turbine blade design method based on three-dimensional euler computations with moving boundaries

A new three-dimensional inverse method for the design of compressor and turbine blades is presented. The method solves the time dependent Euler equations in a numerical domain of which some boundaries (the blade walls) are iteratively modified until a prescribed pressure distribution is reached. Each iteration of the procedure starts with a blade modification, based on the transpiration model and the permeable wall concept. After generating a new mesh the flow field is updated by performing one finite volume time iteration, taking into account the mesh points movement during the time stepping. The blade modifications and time marching computation converge simultaneously to the required geometry and steady-state flow solution. The method makes use of a high resolution three-dimensional finite volume Euler solver, with an upwind-biased evaluation of the advective fluxes for sharp shock wave capturing and low numerical entropy generation. The wall boundary conditions respect the hyperbolic character of the time dependent Euler equations. Non-reflecting boundary conditions are applied along the inlet and outlet boundaries, to favor convergence and limit errors due to shock wave reflection. Applications to turbine and compressor blade design illustrate the capabilities of the method and demonstrate its industrial applicability.

Application of the panel method to subsonic aerodynamic design

Computational methods play an important part in aerodynamic development of modern flying vehicles. In order to improve the flying efficiency many assistant devices, such as leading/trailing flaps are applied. Although there exist many flow analysis methods useful in such aerodynamic configurations, the inverse methods can significantly improve the process of aerodynamic design and final aerodynamic efficiency. This paper presents a panel method which allows for design of 3-dimensional configurations with prescribed pressure distribution at various design angles of attack, including the effect of flap deflections. The high order panel method is applied to analyze the flow. The flap deflections are simulated by changing Neumann boundary conditions. The inverse problem is solved via optimization: the geometry which minimizes differences between design and actual pressure distributions is found iteratively using an optimization technique. The geometrical constrains and regularity conditions can be specified via penalty function concept. The idea of extending the range of applicability of the inverse panel method to transonic flow is also presented.

A two-dimensional Navier—Stokes inverse solver for compressor and turbine blade design

A two-dimensional viscous inverse method for the design of compressor and turbine blades is presented. It iteratively modifies an initial geometry until a prescribed pressure distribution is reached on the blade surface. The method solves the time-dependent Navier—Stokes equations in a numerical domain of which some boundaries (the blade walls) move during the transient part of the computation. The geometry modification algorithm is based on the transpiration principle: a normal velocity distribution is computed from the difference between the actual and prescribed pressure distributions, and is used to modify the blade shape. A time iteration is then performed on this new blade shape, taking into account the grid movement during the time stepping. A two-dimensional upwind finite-volume Navier—Stokes solver has been developed. The multiblock strategy allows for a selective concentration of the discretization points in the zones of higher gradients. Applications to turbine and compressor blade design illustrate the accuracy of the flow computation, the capabilities and efficiency of the inverse method.

A Surface Panel Method for Design of Hydrofoils

A surface panel method treating a boundary-value problem of the Dirichlet type is presented to design a hydrofoil corresponding to a prescribed pressure distribution. An integral equation is derived from Green's theorem, giving a relation between the total potential of known strength and the unknown local flux. Upon discretization, a system of linear simultaneous equations is formed and solved for an assumed geometry. The pseudo local flux, present due to the incorrect positioning of the assumed geometry, plays a role of the geometry corrector, with which the new geometry is computed for the next iteration. Sample designs for a series of pressure distributions of interest are performed to demonstrate the fast convergence, effectiveness and robustness of the procedure. The method is shown equally applicable to designing two- and three-dimensional hydrofoil geometry.

Effect of nose shape on three-dimensional streamlines and heating rates

A new method for calculating the three-dimensional inviscid surface streamlines and streamline metrics using Cartesian coordinates and time as the independent variable of integration has been developed. The technique calculates the streamline from a specified point on the body to a point near the stagnation point by using a prescribed pressure distribution in the Euler equations. The differential equations, which are singular at the stagnation point, are of the two-point boundary-value-problem type. Laminar heating rates are calculated using the axisymmetric analog concept for three-dimensional boundary layers and approximate solutions to the axisymmetric boundary-layer equations. Results for elliptic conic forebody geometries indicate that the location of the point of maximum heating depends on the angle of attack of the conic and the type of conic in the plane of symmetry. This location is in general different from the stagnation point. The new method was found to give smooth predictions of heat transfer in the nose region where previous methods gave oscillatory results. a b by e

Aerodynamic design via optimization

An aerodynamic design optimization method is presented that generates an airfoil, producing a specified surface pressure distribution at a transonic speed. The design procedure is based on the coupled Euler and boundary-layer technology to include the rotational viscous physics which characterizes transonic flows. A leastsquare optimization technique is used to minimize pressure discrepancies between the target and designed airfoils. The method is demonstrated with several examples at transonic speeds. The design optimization process converges quickly, that makes the method attractive for practical engineering applications. I. Introduction I N recent years, computational fluid dynamics (CFD) has become a valuable engineering tool in the aircraft industry. CFD plays a complementary role, not a replacement, to experiments in practical design communities. Rubbert1 showed some good examples of the use of CFD and experiment, in combination, for transonic design. A major strength of CFD is the ability to produce detailed insights into complex flow phenomena. The process of decomposition and parameterization can help identify the cause of weak aerodynamic performance, and the microscopic understanding of the flow can lead to improved design. Continuing advances in computer hardware and simulation techniques provide an unprecedented opportunity for CFD. Now simulations of more complete configurations with more complex physics can be performed at an affordable cost. Accuracy and reliability of the computation have been continuously improved. The use of high-level flow models and large-size refined grids enables one to analyze flows with complicated structures and various length scales. Compared to the remarkable advances in analysis capability, however, relatively few advances have been made in design technology. Conventional design practices, therefore, often depend on analysis methods through iterative cut-and-try approaches. A unique advantage of CFD is the capability of inverse design. Inverse design directly determines the airfoil geometry that produces the pressure distribution specified by a designer. Many existing inverse design methods are based on the potential flow assumption due to its simplicity. Volpe and Melnik2 employed an inverse design method using the nonlinear full potential formulation. Bauer and colleagues3 used the hodograph method that solves the full potential equation in the hodograph plane where the equations are linear. The potential flow model, however, cannot properly represent transonic features such as embedded shock waves and shock-boundarylayer interactions. An accurate analytic capability is a prerequisite for a successful design, because the quality of the design depends on the quality of the method used to predict the flowfield. Several inverse design methods were demonstrated using the Euler formulations by Giles and Drela,4 and Mani.5 Instead of achieving the prescribed pressure distribution, some design methods use a constrained optimization process