Analysis and Inverse Design using Computational Fluid …Analysis and Inverse Design using...

International Journal of Modern Trends in Engineering and Research

www.ijmter.com e-ISSN No.:2349-9745, Date: 28-30 April, 2016

@IJMTER-2016, All rights Reserved 1016

Analysis and Inverse Design using Computational Fluid Dynamics Raju Kumar1, Avinash Kumar2, Dr. K. B. Waghulde3, H.D.Chaudhari4

1,2Students, Department of Mechanical Engineering, J. T. Mahajan C.O.E., Faizpur, Maharashtra, India

3,4Asso. Professor, Department of Mechanical Engineering, J. T. Mahajan C.O.E., Faizpur, Maharashtra, India

[email protected], [email protected], [email protected]

Abstract -- Computational fluid dynamics (CFD) is a simulation tool, which uses powerful computer and applied mathematics to model fluid flow situations for the prediction of heat, mass and momentum transfer and optimal design in industrial processes.The sleek & beautiful aircraft roles down the run away, takes off & rapidly climbs out of sight within a minute, this same aircraft has accelerated to hypersonic speed; still within atmosphere. Its powerful supersonic engine continues to propel aircraft with velocity near 26000 ft/s orbital velocity and vehicle simply coasts into low earth orbit. The physical aspects of any fluid flow are governed by the fundamental principle as Newton’s second law (Navier-Stokes equations). Keywords- Gaussian surface, Grid generation, Flux, Lift and Drag.

I. INTRODUCTION “There are two kinds of persons in this world, the reasonable one and the unreasonable one. The reasonable person adapts himself to the surrounding. The unreasonable one adapts the surrounding to suit him. So eventually all the progress depends on the unreasonable person.” -George Bernard Shaw. Computational Fluid Dynamics (CFD) has grown from a mathematical curiosity to become an essential tool in almost every branch of fluid dynamics, from aerospace propulsion to weather prediction. CFD is commonly accepted as referring to the broad topic encompassing the numerical solution, by computational methods, of the governing equations which describe fluid flow, the set of the Navier-Stokes equations, continuity and any additional conservation equations, for example energy or species concentrations. The attraction of the subject is twofold. Firstly, the desire to be able to model physical fluid phenomena that cannot be easily simulated or measured with a physical experiment, for example weather systems or hypersonic aerospace vehicles. Secondly, the desire to be able to investigate physical fluid systems more cost effectively and more rapidly than with experimental procedures [1]. In design and development, CFD programs are now considered to be standard numerical tools, widely utilised within industry.

Figure 1: Concept of CFD.

International Journal of Modern Trends in Engineering and Research (IJMTER) Volume 3, Issue 4, [April 2016] Special Issue of ICRTET’2016


1.1 CFD Components Human being who inspects and interprets their results. Scientific knowledge expressed mathematically Software which expresses problem in scientific terms Hardware which performs calculations dictated by the software and

II. COMPLETE SYSTEM OF THE NAVIER-STOKES EQUATIONS

In the previous sections, we have separately derived the conservation laws of mass, momentum and energy. For this purpose, we go back to the general conservation law for a vector quantity, which is expressed by Eq. (1). For reasons to be explained later, we will introduce two flux vectors, namely →

and→. The first one → is related to the convective transport of quantities in the fluid [2]. It is usually

termed vector of convective fluxes, although for the momentum and the energy equation it also includes the pressure terms pn Eq. (2) and p (v.n) Eq. (3), respectively. With all this in mind and conducting the scalar product with the unit normal vector n, we can cast Eq. (2) together with Equations (3), (4) and (5) into

∫ →푑Ω + ∮ (→−→)푑푠 = ∫ →푑Ω (1)

The vector of the so-called conservative variables → consists in three dimensions of the following five components

→ =

⎣⎢⎢⎢⎡휌휌푢휌푣휌푤휌퐸⎦

⎥⎥⎥⎤ (2)

For the vector of convective fluxes we obtain

→=

⎣⎢⎢⎢⎡

휌푉휌푢푉 + 푛 푝휌푣푉 + 푛 푝휌푤푉 + 푛 푝

휌퐻푉 ⎦⎥⎥⎥⎤

(3)

With the contra variant velocity V - the velocity normal to the surface element dS - being defined as the scalar product of the velocity vector and the unit normal vector, i.e.,

v≡→.→ = 푛 푢 + 푛 푣 + 푛 푊 (4)

Governing Equations

The total enthalpy H is given by the formula (5). For the vector of viscous fluxes we have with Eq. (4)



→=

⎣⎢⎢⎢⎡푛 푇 + 푛 푇 + 푛 푇푛 푇 + 푛 푇 + 푛 푇푛 푇 + 푛 푇 + 푛 푇푛 휃 + 푛 휃 + 푛 휃 ⎦

⎥⎥⎥⎤ (5)

Where, 휃 = 푢푇 + 푣푇 + 푤푇 + 푘

휃 = 푢푇 + 푣푇 + 푤푇 + 푘휕푇휕푦

휃 = 푢푇 + 푣푇 + 푤푇 + 푘휕푇휕푧

are the terms describing the work of viscous stresses and the heat conduction in the fluid. Finally, the source term reads

→=

⎣⎢⎢⎢⎢⎡

0휌푓 , 푥휌푓 , 푦휌푓 , 푧

휌 →.→ +푞̇ℎ⎦⎥⎥⎥⎥⎤

(6)

In the case of a Newtonian fluid, they describe the exchange (flux) of mass, momentum and energy through the boundary dR of a control volume R, which is fixed in space. Applying Gauss's theorem, Eq. (1) can be re-written in differential form. In some instances, for example in turbo-machinery applications or geophysics, the control volume is rotating (usually steadily) about some axis. In such a case, the Navier-Stokes equations are transformed into a rotating frame of reference. As a consequence, the source term Q has to be extended by the effects due to the Coriolis and the centrifugal force [3]. In other cases, the control volume can be subject to translation or deformation. This happens, for instance, when fluid-structure interaction is investigated. Then the Navier- Stokes equations have to be extended by a term, which describes the relative motion of the surface element dS with respect to the fixed coordinate system.

Figure 2 : Helicopter Airflow



The lift for helicopter can be derived from Eq. (2) of the Navier-Stokes Equations in a similar way (as for the aeroplane) with the role of ux and uz interchanged because the airflow pattern is different. Thus for uz>>ux, Eq. (2) becomes:

uz [d(uz)/dz] = - (1/ρ ) dp/dz (7)

Integrating Eq. (7) yields:

{ρ [(-uz2)2 - (-uz1)2]} / 2 = (p2 – p1) (8)

where uz is negative as it is pointing toward the negative z direction. It has a form similar to Eq. (8) for the aeroplane. Computation for the lifting force follows exactly the same line as developed previously.

III. GRID GENERATION

In CFD, the domain is divided in to some shapes, such as triangles or quadrilaterals; and assumption is made that flows properties are constant within individual cell, but they do vary cell to cell. This task is called as great generation [4]. The set of cells are called grids or meshes.

Grids must satisfy the following properties:

I Cells must be simple in shape (say triangle or rectangle in 2D) II Cells must completely cover the given domain, III No two cells must overlap, IV Cells must convex in shape.

Two general approaches are available to deal with complex geometries;

(i) STRUCTURED GRID.

(ii) UNSTRUCTURED GRID.

3.1 Structured Grid In structured three dimensional grid, one can associate with each computational cell an ordered triple of indices (i,j,k).Thus if Ni, Nj, Nk are number of cells in i, j, k, index directions respectively, then the number of cells in entire mesh is Ni * Nj * Nk. Additionally it is seen that each interior vertex is sheared by eight neighbouring cells.

Figure 3 : Structured Grid



3.1.1 O-Grid Another commonly used grid generation method for airfoils and other external flow (where the fluid flows around the outside of a solid body as opposed to inside it) is known as an O-grid. This method involves specifying the computational points on the surface and allowing the grid to grow out and away from the surface.

Figure 4 : O-Grid with Triangular Domain (Magnified)

3.1.2 H-Grid The H-grid is one of the simpler grids to generate. It is useful for flows that can be contained roughly within a four-sided container. These are usually internal flows where the fluid is moving within solid walls. The following is an example of an H-grid describing the geometry for flow within a duct with a contraction as shown in Figure. (5) and Figure. (6).

Figure 5 : H-GRID (Grid Generation)

Figure 6 : H-GRID (Simulation)

3.2 Unstructured Grid Another method of grid generation, called unstructured grids, involves a triangular layout of the computational points. There are several advantages of using unstructured grids. One advantage is that the technique tends to work better on complex geometries. This is because if all the grid points were shown simultaneously it would be impossible to distinguish the flow field in question as shown in Figure 7. A popular type of unstructured grid consists of tetrahedral element.



Figure 7 : Unstructured grid

These grids tend to be easier to generate than those composed of hexahedral elements, but they generally have poorer numerical accuracy.

IV. SIMULATION

The computing times for a flow simulation depend on;

The choice of numerical algorithms and data structures Linear algebra tools, stopping criteria for iterative solvers Discretization parameters (mesh quality, mesh size, time step) Cost per time step and convergence rates for outer iterations Programming language (most CFD codes are written in FORTRAN) Many other things (hardware, vectorization, parallelization etc.)

Advances in airfoil shape are constantly being made and modern CFD techniques have made the process much faster [5]. This eliminates the need for physically testing every candidate and focuses on the best potential ones as shown in Figure 8.

Figure 8 : Simulation of Airfoils (SUBSONIC SPEED)

The blue "cool" colours indicate low speed flow and the red "hot" indicate high-speed flow. As expected, the upper surface shows higher speed flow, which provides the low-pressure lift for an airfoil. The wake also indicates low pressure that contributes to the airfoil drag as shown in Figure 9.



Figure 9 : Simulation of Airfoils (TRANSONIC SPEED)

This image is of an airfoil at transonic speed, meaning that both subsonic and supersonic flow is present about the airfoil [6]. This usually happens when the flow decelerates from supersonic to subsonic speed.

V. APPLICATIONS

5.1 Aerospace Applications Today, CFD is used to calculate three dimensional flow fields over real airplanes. Such complete flow field calculations over entire airplane configurations is a major step in enhancing the overall airplane design process.

Figure 10 : The image shows the pressure contours on the upper surface of a wing.

CFD results can be used to optimize the variation of airfoil size and shape at different sections of the wing in order to insure that the wing is producing lift efficiently and also maintains structural stability [7].

5.2 High-Speed Vehicles Computational simulations are used frequently in the development of high-speed vehicles such as rockets, spacecraft and hypersonic aircraft. Hot exhaust gas of the engines would be very expensive to simulate in a wind tunnel test, and so CFD computations are the only practical way of getting results for complex flows.



Figure 11 : Simulation of aircraft

The above image Figure. (11) is a simulation of a hypersonic aircraft. Solutions on the surface and at particular slices away from the aircraft are shown. These aircraft possible design in a wind tunnel would be prohibitively expensive.

5.3 Lift and Drag of Aircraft

Figure 12 : Lift and Drag of Aircraft in Accordance to Angle of Attack

Ontogenetic trends wing performance at in vivo Reynolds numbers: (i) coefficient of lift (CL) versus angle of attack (α); (ii) coefficient of drag (CD) versus α; (iii) CL versus CD; and (iv) CL/CD versus α. In vivo angles of attack (α) are indicated by shaded areas; maximum L:D ratios are indicated by squares in (iii).

CONCLUSION

CFD is a new “third dimension” in fluid dynamics, equally sharing the stage with the other dimensions of pure theory and pure experiments i.e. CFD enables the user to get a 3-dimensional virtual reality not only visually but supported by computations which otherwise would be very complex to solve. It has reduced research time to nearly about one-fourth of previous time. Also it allow to study the various aspects of very complex and huge applications like aero plane, rocket, submarine etc on computer without physically testing them in wind tunnel (which is impossible).



Hence it save millions of rupees in design aspect and also increases the performance of the various fluid related appliances. In future, with more efficient and developed computer technology CFD will become the basic necessity of industries.

REFERENCE

[1]Aftosmis, M.J., Berger, M.J., Melton, J.E., 1997. Robust and efficient cartesian mesh generation for component-

based geometry. Paper presented at 35th AIAA Aerospace Sciences Meeting, Reno, Nevada, USA. [2]Radespiel, R.: A Cell- Vertex Multigrid Method for the Navier-Stokes Equations. NASA TM-101557, 1989. [3]Haselbacher, A.; Blazek, J.: On the Accurate and Efficient Discretisation of the Navier-Stokes Equations on Mixed

Grids. AIAA Paper 99-3363, 1999; also AIAA Journal, 38 (2000), pp. 2094-2102. [4]Upender K. Kaul, (2003) New boundary constraints for elliptic systems used in grid generation problems. Journal of

Computational Physics 189, 476-492. [5]Engquist, B.; Majda, A.: Absorbing Boundary Conditions for Numerical Simulation of Waves. Mathematics of

Computations, 31 (1977), pp. 629-651. [6]Drela, Mark, "XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils," in Springer-Verlag

Lecture Notes in Engineering, No. 54, 1989. [7]Maskew, Brian, “Program VSAERO Theory Document: A Computer Program for Calculating Nonlinear

Aerodynamic Characteristics of Arbitrary Configurations”, NASA CR-4023, 1987.

Analysis and Inverse Design using Computational Fluid …Analysis and Inverse Design using...

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