scholarly journals Comparison of Linear Vortex Panel Method and Finite Volume Method for Calculation of Generated Lift in Potential Flow Over Two-Dimensional Car Bodies

2020 ◽  
Author(s):  
Saeid Moammaei ◽  
Mehran Khaki Jamei ◽  
Morteza Abbasi
Keyword(s):  
Pressure Distribution ◽  
Lift Coefficient ◽  
Control Point ◽  
Panel Method ◽  
The Body ◽  
Aerodynamic Load ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Impact

Abstract This paper describes one of the aspects of the panel method to analyze the aerodynamic characteristics of a sedan. The linear vortex panel method has been developed to simulate the ideal flow over a two-dimensional arbitrary car and, it also calculates the aerodynamic load on the body. By satisfying the boundary conditions on each control point, our linear algebraic equations are obtained. The results are sensitive to the distribution of the panels over the body thus the body is broken up equally into very small panels. After solving the set of equations, the vortices strength is obtained and the pressure distribution for the upper and the lower surface of the body is calculated. The impact of the angle of attack on the aerodynamic behavior of the intended car is investigated and it is found that the lift coefficient increases with the free stream angle from -4 to 4. The accuracy of the results has been determined by checking them against the standard CFD data. The pressure distribution trend is found very much in confirmation with the CFD results, however, a discrepancy at the rear end is observed. Therefore, it can be concluded that this method does not seem practical for geometries with steep slopes in the rear part of the car. Finally, both methods are applied to the other modified geometries with lower slopes at the rear section and the results compare well with the fluent.

A Linearized Two-Dimensional Theory for High-Speed Hydrofoils Near the Free Surface

1966 ◽  
Vol 10 (01) ◽  
pp. 25-48
Author(s):  
Richard P. Bernicker
Keyword(s):  
Direct Problem ◽  
High Speed ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Pressure Loading ◽  
Dimensional Theory ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sectional Shape ◽  
Infinite Set

A linearized two-dimensional theory is presented for high-speed hydrofoils near the free surface. The "direct" problem (hydrofoil shape specified) is attacked by replacing the actual foil with vortex and source sheets. The resulting integral equation for the strength of the singularity distribution is recast into an infinite set of linear algebraic equations relating the unknown constants in a Glauert-type vorticity expansion to the boundary condition on the foil. The solution is achieved using a matrix inversion technique and it is found that the matrix relating the known and unknown constants is a function of depth of submergence alone. Inversion of this matrix at each depth allows the vorticity constants to be calculated for any arbitrary foil section by matrix multiplication. The inverted matrices have been calculated for several depth-to-chord ratios and are presented herein. Several examples for specific camber and thickness distributions are given, and results indicate significant effects in the force characteristics at depths less than one chord. In particular, thickness effects cause a loss of lift at shallow submergences which may be an appreciable percentage of the total design lift. The second part treats the "indirect" problem of designing a hydrofoil sectional shape at a given depth to achieve a specified pressure loading. Similar to the "direct" problem treated in the first part, integral equations are derived for the camber and thickness functions by replacing the actual foil by vortex and source sheets. The solution is obtained by recasting these equations into an infinite set of linear algebraic equations relating the constants in a series expansion of the foil geometry to the known pressure boundary conditions. The matrix relating the known and unknown constants is, again, a function of the depth of submergence alone, and inversion techniques allow the sectional shape to be determined for arbitrary design pressure distributions. Several examples indicate the procedure and results are presented for the change in sectional shape for a given pressure loading as the depth of submergence of the foil is decreased.

Function Airfoil and its Pressure Distribution and Lift Coefficient Calculation

2013 ◽  
Vol 328 ◽  
pp. 351-356 ◽  
Author(s):  
Hai Bo Jiang ◽  
Zhong Qing Cheng ◽  
Yun Peng Zhao
Keyword(s):  
Pressure Distribution ◽  
Distribution Curve ◽  
Integration Method ◽  
Lift Coefficient ◽  
Analytical Formula ◽  
Leading Edge ◽  
Independent Variable ◽  
The One ◽  
The Impact ◽  
The Relationship

To study the impact of an airfoil shape on performance, a curve expression of airfoil shape was proposed, the analytical formula for the pressure distribution of flow around the airfoil was derived, and the pressure distribution view around airfoil with azimuth as the independent variable was put forward, which can clearly express the details of the pressure distribution curve on airfoil leading edge. Used both the pressure distribution integration method and Blasius theorem, the lift coefficient calculation formulas of ideal fluid flow around the airfoil were derived respectively, and the same results were obtained. Studies have shown that the shape of an airfoil can be expressed by a function, and various types of shapes can be easily obtained by adjusting the constant value in the expression; The pressure distribution and lift coefficient can be calculated by analytical method; For function airfoil, lift coefficient formula could be derived by two methods, and could be verified with each other. The one-to-one relationship exists between the constant values in the airfoil function, airfoil shapes and airfoil performances, and the relationship expression was given in this paper.

scholarly journals Numerical Solution for the 2D Linear Fredholm Functional Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Neda Khaksari ◽  
Mahmoud Paripour ◽  
Nasrin Karamikabir
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Unknown Function ◽  
Numerical Solutions ◽  
Functional Integral ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

In this work, a numerical method is applied for obtaining numerical solutions of Fredholm two-dimensional functional linear integral equations based on the radial basis function (RBF). To find the approximate solutions of these types of equations, first, we approximate the unknown function as a finite series in terms of basic functions. Then, by using the proposed method, we give a formula for determining the unknown function. Using this formula, we obtain a numerical method for solving Fredholm two-dimensional functional linear integral equations. Using the proposed method, we get a system of linear algebraic equations which are solved by an iteration method. In the end, the accuracy and applicability of the proposed method are shown through some numerical applications.

Methodology to Evaluate Slamming Loads on Ocean Platforms

2008 ◽  
Author(s):  
Marcio Domingues Maia Junior ◽  
Antonio Carlos Fernandes ◽  
Marcela Trindade ◽  
Andre Ramiro
Keyword(s):  
Free Surface ◽  
Pressure Distribution ◽  
Pressure Sensor ◽  
Potential Flow ◽  
Experimental Methods ◽  
The Body ◽  
Platform Motion ◽  
Air Pocket ◽  
Accuracy And Precision ◽  
The Impact

The purpose of the study is suggest a methodology to be applied in ocean platforms and ships in order to appraise the maximum impact pressure due to the slamming occurrence in the hull shape near its bottom or horizontal regions. This methodology uses a theory based on potential flow. However, there are some phenomena such as creation of a compressible air pocket between the body and free surface at the impact moment that requires a more complete theory and or experimental methods. This gives rise to experimental coefficients to reduce the theoretical errors. The procedure presented here goes by the platform motion dynamics and “impact topology” to allow the potential to be used. Due to the complexity of the phenomenon studied and need for certifying accuracy and precision of the results, tank tests at the LabOceano model basin were carried out. The results showed a good fitting between numerical results and experiments. It should also be pointed out that the pressure sensor used in these experiments gives a pressure distribution over the instrumented area what brings more reliability on the results and a better visibility to the slamming phenomenon. Lastly the methodology in this work stands out as an important tool to evaluate slamming loads.

Mathematical model of piston-tool interaction in rock fracture by impact

2020 ◽  
pp. 94-103
Author(s):  
A. B. Zhabin ◽  
I. M. Lavit ◽  
A. V. Polyakov ◽  
Z. E. Kerimov
Keyword(s):  
Finite Difference ◽  
Boundary Value Problems ◽  
Variational Equation ◽  
Rock Fracture ◽  
Boundary Value ◽  
Calculation Error ◽  
Algebraic Equations ◽  
Time Curve ◽  
Linear Algebraic Equations ◽  
The Impact

The authors justify a computation model of a machine percussion system simulated by elastic cylindrical rods subjected to maximal axial load at minimal strain. Rock mass is assumed as a perfectly solid block. The piston and tool are described by the values of length, cross-section area, density and Young’s modulus. The model determines the force applied by the tool on the rock as function of time. It is assumed that transverse displacements and velocities of the rods are negligeable as compared with the axial displacements and velocities, while the rods are free from the action of the external forces different from the restraining forces. The variational equation expresses the principle of possible displacements. The variations are independent of time. The initial and boundary conditions are considered. The variational equation is solved using the method of straight lines, with replacement of a time differentiation operator by the finite difference operator. The problem reduces to the successive solving of boundary value problems with variable right-hand sides. The finite difference scheme is the approved implicit scheme of Crank-Nicolson. The boundary value problems are solved using the finite element method at each step of integrating. As a result, the variational equation transforms into a system of linear algebraic equations, and the reduced solution of this system yields the wanted force. The calculations are illustrated by the tool press force-time curve plotted with a step of 0.1 µs for hydropercussion machine G100 by Rammer, Finland. The relative calculation error of the impact duration and maximal force (in absolute magnitude) is not higher than 0.1%.

Calculation of Nonlifting Potential Flow About Arbitrary Three-Dimensional Bodies

1964 ◽  
Vol 8 (04) ◽  
pp. 22-44 ◽  
Author(s):  
John L. Hess ◽  
A. M. O. Smith
Keyword(s):  
Body Surface ◽  
Potential Flow ◽  
Fluid Velocity ◽  
Normal Component ◽  
Three Dimensional ◽  
Analytic Solutions ◽  
The Body ◽  
Algebraic Equations ◽  
Source Density ◽  
Linear Algebraic Equations

A general method is described for calculating, with the aid of an electronic computer, the incompressible potential flow about arbitrary, nonlifting, three-dimensional bodies. The method utilizes a source density distribution on the surface of the body and solves for the distribution necessary to make the normal component of fluid velocity zero on the boundary. Plane quadrilateral surface elements are used to approximate the body surface, and the integral equation for the source density is replaced by a set of linear algebraic equations for the values of the source density on the quadrilateral elements. When this set of equations has been solved, the flow velocity both on and off the body surface is calculated. After the basic ideas and equations have been derived end discussed, the accuracy of the method is exhibited by means of comparisons with analytic solutions, and its usefulness is shown by comparing calculated pressure distributions with experimental data. Some of the design problems to which the method has been applied are also presented, to indicate the variety of flow situations that can be calculated by this approach.

Hydrodynamic Trends for Preliminary Design of Fully Cavitating Hydrofoil Sections

1977 ◽  
Vol 14 (01) ◽  
pp. 70-85
Author(s):  
Blaine R. Parkin ◽  
Robert F. Davis ◽  
Joseph Fernandez
Keyword(s):  
Pressure Distribution ◽  
Lift Coefficient ◽  
Flow Theory ◽  
Cavity Flow ◽  
Cavitation Number ◽  
Viscous Drag ◽  
Preliminary Design ◽  
Two Dimensional ◽  
Cavity Thickness ◽  
Cavitating Hydrofoil

The object of this numerical study is to consider possible hydrodynamic trends for use in trade-off studies for the preliminary design of fully cavitating hydrofoil sections. Hydrodynamic data are obtained from inverse calculations which are based upon two-dimensional linearized cavity-flow theory. Supplementary data are also calculated from the direct problem of linearized cavity-flow theory in order to show off-design performance trends and to assess the effects of cavity-foil interference on the operating range of selected profiles. For the inverse calculations one specifies design values of the lift coefficient, cavitation number, and cavity thickness at the trailing edge, as well as the shape of the pressure distribution on the wetted surface of the hydrofoil section. In accordance with this specification, the ordinates of the profile wetted surface and upper-cavity contour are calculated, together with values of drag coefficient, moment coefficient, and attack angle at the design point. The paper summarizes the results of a parametric study of the effects of design cavitation number, lift coefficient, cavity thickness, and pressure distribution shape upon hydrofoil section performance and geometry. Three-dimensional wing effects, viscous drag, and the effects of structural design criteria are all outside the scope of the study. Results pertaining to steady two-dimensional cavity flows of an ideal incompressible fluid past a rigid hydrofoil section are presented.

Pressure distribution and mass injection effects in the transitional separated flow over a spiked body at supersonic speed

1966 ◽  
Vol 24 (2) ◽  
pp. 209-223 ◽  
Author(s):  
Mansop Hahn
Keyword(s):  
Pressure Distribution ◽  
Separated Flow ◽  
Injection Rate ◽  
Pressure Variation ◽  
The Body ◽  
Air Injection ◽  
Two Dimensional ◽  
Supersonic Speed ◽  
Schlieren Technique ◽  
Mass Injection

Pressure distribution and the effect of air injection in the separated flow over a spiked-hemisphere were investigated at a Mach number of 3·3, and Reynolds number around the transitional value. Pressure distribution along the spike as well as over the body was measured in the absence of injection. Air was injected into the separated flow at the spike tip and base and reattachment region through one or more orifices drilled normal to the surface, and the resulting flow patterns were observed using the schlieren technique. The results show that (i) the pressure variation along the spike is similar to a two-dimensional separated flow in the transition régime; and (ii) the mass injection at the spike tip has a strong destabilizing effect regardless of injection rate, while the injection from spike base and reattachment region can be either slightly stabilizing or destabilizing depending on the flow condition.

The motion of a rigid body in viscous fluid bounded by a plane wall

1989 ◽  
Vol 207 ◽  
pp. 29-72 ◽  
Author(s):  
Richard Hsu ◽  
Peter Ganatos
Keyword(s):  
Aspect Ratio ◽  
Stokes Equations ◽  
Hydrodynamic Force ◽  
Orientation Angle ◽  
Boundary Integral ◽  
The Body ◽  
Boundary Integral Method ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Planar Wall

The boundary-integral method is used to calculate the hydrodynamic force and torque on an arbitrary body of revolution whose axis of symmetry is oriented at an arbitrary angle relative to a planar wall in the zero-Reynolds-number limit. The singular solution of the Stokes equations in the presence of a planar wall is used to formulate the integral equations, which are then reduced to a system of linear algebraic equations by satisfying the no-slip boundary conditions on the body surface using the boundary collocation method or weighted residual technique.Numerical tests for the special case of a sphere moving parallel or perpendicular to a planar wall show that the present theory is accurate to at least three significant figures when compared with the exact solutions for gap widths as small as only one-tenth of the particle radius. Higher accuracy can be achieved and solutions can be obtained for smaller gap widths at the expense of more computation time and larger storage requirements.The hydrodynamic force and torque on a spheroid with varying aspect ratio and orientation angle relative to the planar wall are obtained. The theory is also applied to study the motion of a toroidal particle or biconcave shaped disc adjacent to a planar wall. The coincidence of the drag and torque of a biconcave-shaped body and a torus having an aspect ratio b/a = 2 with the same surface area shows that in this case the hole of a torus has little influence on the flow field. On the other hand, for an aspect ratio b/a = 10, the effect of the hole is significant. It is also shown that when the body is not very close to the wall, an oblate spheroid can be used as a good approximation of a biconcave-shaped disc.

A Two-Dimensional Numerical Investigation of the Hysteresis Effect on Vortex Induced Vibration on an Elastically Mounted Rigid Cylinder

2011 ◽  
Vol 134 (2) ◽  
Author(s):  
Juan B. V. Wanderley ◽  
Sergio H. Sphaier ◽  
Carlos Levi
Keyword(s):  
Vortex Shedding ◽  
Stokes Equations ◽  
Lift Coefficient ◽  
Practical Interest ◽  
Numerical Results ◽  
Hysteresis Effect ◽  
The Body ◽  
Curvilinear Coordinates ◽  
Two Dimensional ◽  
Vortex Induced Vibration

The hysteresis effect on the vortex induced vibration (VIV) on a circular cylinder is investigated by the numerical solution of the two-dimensional Reynolds averaged Navier-Stokes equations. An upwind and total variation diminishing (TVD) conservative scheme is used to solve the governing equations written in curvilinear coordinates and the k-ɛ turbulence model is used to simulate the turbulent flow in the wake of the body. The cylinder is supported by a spring and a damper and free to vibrate in the transverse direction. In previous work, numerical results for the amplitude of oscillation and vortex shedding frequency were compared to experimental data obtained from the literature to validate the code for VIV simulations. In the present work, results of practical interest are presented for the power absorbed by the system, phase angle, amplitude, frequency, and lift coefficient. The numerical results indicate that the hysteresis effect is observed only when the frequency of vortex shedding gets closer to the natural frequency of the structure in air.

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