Ball-spine Algorithm Research Articles

Optimization of the target pressure distribution and inverse design of an axial turbine blade

The most critical step in designing a turbomachine is to design the blades. In this study, an axial turbine blade is designed using an inverse design method called Ball-Spine Algorithm. In some cases, the target pressure distribution is not specified in the inverse design process, and it has to be determined arbitrarily by the designer. This study is conducted in order to compute the optimal target pressure distribution by maximizing the cascade blade efficiency. Optimization is carried out using the Genetic Algorithm. Two commercial software packages, ANSYS and MATLAB, are used in this research. Since inverse design is an iterative process, these software packages are linked together to execute the whole procedure automatically. Validation of the approach is accomplished by redesigning an existing blade. Then the optimization method is applied, and the desired blade is inversely designed. Results show that the cascade blade efficiency increases by about 9% on average for this blade.

Quasi-3D inverse design of S-shaped diffuser with specified cross-section distribution; super-ellipse, egg-shaped, and ellipse

Recently, an inverse design method called the ball spine algorithm (BSA) is introduced to design S-duct diffusers with elliptic cross-sections. The technique is developed for the 3-D design of S-shaped ducts with special cross-sectional profiles in the present work. The upper and lower lines of the S-duct symmetric section are modified under the BSA. Two types of special cross-section profiles are used to improve the aerodynamic properties of the duct. At the S-duct inlet, the super-ellipse profile is used to minimize the duct inlet height. At the aft part of the duct, the egg-shaped profile is predefined to decrease the pressure difference between the upper and lower lines and to diminish the adverse pressure gradient along the upper line. Finally, the flow through the designed S-duct is numerically analyzed using a viscous flow solver with the SST turbulence model to validate the designed S-duct performance.

Inverse design of 3D curved ducts using a 3D-upgraded ball-spine algorithm

ABSTRACT Achieving a unique solution for the 3D inverse design of a curved duct is a challenging problem in aerodynamic design. The centre-line curvature, and cross-sections’ area and shape of a 3D curved duct influence the wall pressure distribution. All the previous developments on the ball-spine method were limited to 2D and quasi-3D ducts, in which only the upper and lower lines of the symmetry plane were modified based on the target pressure distribution. In the present work, the ball-spine method was three-dimensionally developed for the design of curved ducts while considering the effects of cross-sectional shape and area. To validate the method, all the nodes of a 3D duct wall were iteratively corrected under the modified ball-spine method to reach the target geometry. The effects of the ball movement directions (spines) and the grid generation scheme in achieving the unique solution in inverse design were evaluated. The results showed that the new method converges to a unique solution only if the streamline-based grids are applied for the flow numerical solution, and the horizontal spines are considered as the directions for the displacement of the nodes. Finally, the wall pressure distribution of a high-deflected 3D S-shaped diffuser was three-dimensionally modified to reduce the separation, secondary flow, and flow distortion.

Determining the Optimal Geometry for Producing Isotherms on a Boundary Surface

A shape-design problem is examined in this study that consists of determining the optimal boundary shape of a conductive body with a heating object or heating substrate that will yield a uniform boundary temperature. The Levenberg–Marquardt method and commercial software are used in this shape-design algorithm. The validity of the design analysis is verified using numerical experiments. Without considering the constraint of the domain area, different test cases examined previously by Mayeli et al. (“Inverse Shape Design for Heat Conduction Problems via the Ball Spine Algorithm,” Numerical Heat Transfer, Part B: Fundamentals, Vol. 69, No. 3, 2016, pp. 249–269) are reconsidered in this work to justify the validity and superiority of the present algorithm. The estimated results in the present work are then compared with the results given by Mayeli et al. It is found that the difficulty in choosing the best value for the overall underrelaxation factor reported by Mayeli et al. can be avoided by using the present algorithm, and it needs fewer iterations than reported by Mayeli et al. Next, when the constraint of the domain area is considered, the numerical experiments reveal that the optimal boundary shape with uniform temperature requirement can always be obtained.

Surface Shape Design in Different Convection Heat Transfer Problems Via a Novel Coupled Algorithm

In this study, a new coupled surface shape design (SSD) methodology named direct design method is presented for the solution of problems containing different types of convection heat transfer in which a specific distribution of either heat flux or temperature is given instead of the shape of a boundary. In the proposed method, the governing equation, without using any mathematical transformation for the physical domains, is manipulated so that the grid generation, solving fluid flow, and heat transfer as well as shape updating can all be carried out simultaneously. Five different inverse shape design problems containing different types of convection heat transfer are solved by the proposed method. All the problems are also solved using the ball-spine algorithm (BSA), which is a recently developed de-coupled algorithm, for the sake of comparison. In all problems, the effects of using different under-relaxation parameters are investigated and the capability of both approaches is compared with each other. The results show that the proposed coupled method can solve the problems better than the BSA in the sense that the direct design method converges sooner than the BSA when the same under-relaxation parameter is used for both methods. Also, it is shown that the computational cost of solving a SSD problem using the direct design method is slightly greater than solving an analysis problem.

Development of the ball-spine algorithm for the shape optimization of ducts containing nanofluid

ABSTRACTIn this paper a novel shape design method is introduced for the numerical solution of inverse heat convection problems (IHCPs) of nanofluids. The proposed method is a novel extension of the ball-spine algorithm (BSA) inverse method, which is recently adapted for inverse heat transfer problems (IHTPs). Here it is shown how, by a novel physical-sense remedy, the method is applicable to IHCPs as well. In this respect, after validation of the proposed method, two types of IHCPs are introduced and numerically solved. In both types of introduced inverse problems, the objective is shape optimization of a duct containing such steady-state incompressible laminar nanofluid flow that it satisfies a prescribed heat flux distribution along the walls of the geometry. The results show merits and robustness of the BSA application in capturing the target geometry corresponding to a given heat flux distribution in forced heat convection problems with a low computational cost.

Determination of desired geometry by a novel extension of ball spine algorithm inverse method to conjugate heat transfer problems

Abstract In the present study, numerical solution of an inverse conjugate heat transfer problem including conduction and forced convection is carried out via the ball spine algorithm (BSA). The BSA inverse method is originally developed for inverse shape design of pure fluid flow problems with a desired pressure distribution along an unknown surface but in this paper it is shown how proposing a novel remedy enables the BSA method of solving inverse heat transfer problems as well. The proposed method has a truly low computational cost accompanied by high convergence rate. The inverse problem is defined as heat source surrounded by a solid medium exposed to free stream in external flow. Heat is generated by a heat source and spreads all over the medium by conduction and then is transferred to free stream by forced convection. The objective is finding a configuration for the solid body to satisfy a prescribed uniform temperature along its surface. Solution of the mentioned inverse problem is dependent on different combinations of the dominant parameters including Reynolds number ( Re ), thermal conductivity ratio ( k f / k s ), desired outer surface temperature ( θ s ) and also Prandtl number ( Pr ).

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACTIn this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.

Three-Dimensional Design of Axial Flow Compressor Blades Using the Ball-Spine Algorithm

Recently a new inverse design algorithm has been developed for the design of ducts, called ball-spine (BS). In the BS algorithm, the duct walls are considered as a set of virtual balls that can freely move along some specified directions, called ‘spines’. Initial geometry is guessed and the flow field is analyzed by a flow solver. Comparing the computed pressure distribution (CPD) with the target pressure distribution (TPD), new balls positions for the modified geometry are determined. This procedure is repeated until the target pressure is achieved. In the present work, the ball-spine algorithm is applied to three-dimensional design of axial compressor blades. The design procedure is tested on blades based on NACA65-410 and NACA65-610 profiles and the accuracy of the method is shown to be very good. As an application, the pressure distribution of the blade with NACA65-610 profiles is modified and the pressure gradient in the aft part of the blade is decreased and selected as target pressure distribution. The corresponding geometry which satisfies the target pressure is determined using the BS design algorithm.

Applying the ball-spine algorithm to the design of blunt leading edge airfoils for axial flow compressors

An inverse design algorithm, called the ball-spine (BS) algorithm, is developed for the design of axial flow compressor blades. This algorithm has been previously applied to the design of 2D sharp leading edge airfoils. In the present study, the BS algorithm is extended to the design of blunt leading edge compressor blade sections. The algorithm receives a target pressure loading and an initial guess for its geometry. Through the design procedure, the geometry is evolved until the target pressure is achieved. The accuracy of the method is assessed against two airfoils, namely, NACA65-410 (with the initial guess for geometry as a flat-plate-airfoil) and NACA65-610 (with the initial guess for geometry as NACA65-410). The method is shown to be efficient and accurate, and can capture the target geometry in approximately 50 evolutions. As an application, a new airfoil with 5% more lift than NACA65-610 is designed, for which NACA65-610 geometry is used as the initial guess.

Aerodynamic Design of S-Shaped Diffusers Using Ball–Spine Inverse Design Method

Recently, an inverse design algorithm called ball–spine algorithm (BSA) was introduced for the design of 2D ducts. In this approach, the walls are considered as a set of virtual balls that can move freely along the straight directions called spines. In the present work, the method is developed for quasi-three-dimensional (quasi-3D) design of S-shaped ducts with a predefined width. To do so, the upper and lower lines of the S-duct symmetric section are modified under the BSA and then, the 3D S-duct geometry is obtained based on elliptic cross-sectional profiles. The target pressure distributions (TPDs) along the upper and lower lines are prescribed so that separation does not occur. Finally, the flow through the designed S-duct is numerically analyzed using a viscous flow solver with the SST turbulence model to validate the designed S-duct performance. The performance of the designed S-duct is compared to original and optimized versions of a benchmark S-duct diffuser. Results show that the present S-duct has a better performance.

Optimization of a seven-stage centrifugal compressor by using a quasi-3D inverse design method

This paper focuses on performance improvement of a centrifugal compressor. An inverse design method for 3D design approaches is formulated to address this concern. The design procedure encompasses two major steps. First, with the use of ball spine algorithm, which is an inverse design algorithm, on the meridional plane of impeller, the hub and shroud of impeller are computed based on a modified pressure distribution along them. Second, an original and progressive algorithm is developed for design of blade camber line profile on the blade-to-blade planes of impeller based on blade loading improvement. Full 3D analysis of the current and designed compressor is accomplished by using a Reynolds-averaged Navier-Stokes equations solver. A comparison between the analysis results of the current and designed compressor shows that the total-to-total isentropic efficiency and pressure ratio of the designed compressor under the same operating conditions are enhanced by more than 4.5% and 5%, respectively.

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%.