Prediction of lift characteristics using grid free cfd euler solver

PREDICTION OF LIFT CHARACTERISTICS USING GRID FREE CFD EULER SOLVER Nagabhushana N1, Dr. V.Ramesh2 1,2

Department of Mechanical Engg., New Horizon College of Engineering, Bangalore.

Scientist, NAL, Bangalore

Abstract: The latest developments in the least squares kinetic upwind method (LSKUM), a kinetic theory based grid free approach for the solution of Euler equations. A single step higher order scheme through modified CIR splitting is presented. A new weighted least squares method has been used in the present work which simplifies the 2-D formulae to an equivalent 1-D form. This is achieved through diagonalisation of the least squares matrix through suitable choices of the weights.

A 2-D unsteady Euler code has been developed incorporating all the above ideas along with the well known dual time stepping procedure. The code has been verified and validated for the standard test case AGARD CT(5) which corresponds to unsteady flow past oscillating NACA0012 airfoil pitching about quarter chord. Good comparisons with the experimental values have been obtained.

I. INTRODUCTION The work on the latest developments of a grid free method for computing inviscid unsteady flow past multiple moving bodies. Least squares kinetic upwind method (LSKUM) [4] and [5] is a kinetic theory [3] based grid free scheme for solving the inviscid compressible Euler equations of gas dynamics. This method has also been applied to compute viscous flows [9]. LSKUM has been extended to applications with moving nodes (LSKUM_MN) [11]. Spatially higher order accuracy is achieved (in LSKUM as well as LSKUM_MN) using the two step defect correction method. In case of LSKUM_MN, it has been shown that defect correction step necessitates the recalculation of moving fluxes [11] and [12] at not only all the immediate neighbouring nodes (secondary nodes) but also at the neighbouring points of the secondary nodes. This leads to considerable increase in computational time compared to steady-state computations. In the present work proposed to use the modified CIR splitting (MCIR) [1] and [11] to obtain spatially higher order accuracy in LSKUM_MN. MCIR is a method to achieve spatially higher order accuracy without using the two step defect correction method. Essentially the dissipation term present in the first order scheme is modified to get an equivalent higher order scheme where the dissipation terms are comparable to the usual second-order schemes. This leads to a single step higher order scheme, thereby reducing the computational costs. Apart from the implementation of MCIR in LSKUM_MN, also adopting the weighted least squares approach based on eigenvector basis [7]. In this approach the least squares approximations for all the derivatives reduce to an equivalent 1-D form. This again helps in further reducing the computational time. For the unsteady calculations we have used the well known dual stepping procedure [11]. The present method with all the above-mentioned techniques has been validated for the AGARD [1] CT5 standard test case. This is the case of unsteady transonic flow past an oscillating NACA0012 airfoil. In order to demonstrate the power of the method to handle multiple oscillating bodies, computed flow past an oscillating pair of NACA0012 airfoils, one behind the other.

II. LEAST SQUARES KINETIC UPWIND METHOD ON MOVING NODES Here a brief description of the formula for 2-D LSKUM_MN. Consider the 2-D Boltzmann equation (1)

@IJRTER-2016, All Rights Reserved

113

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457]

where f is the velocity distribution function, v1 and v2 are the Cartesian components of the molecular velocity. J represents a collision term which vanishes in the Euler limit, when f is a Maxwellian distribution, F, which in two dimensions is given by (2)

where β=1/(2RT), Ď is the fluid density, I is internal energy variable, I0 is the internal energy due to non-translational degrees of freedom, and u1 and u2 are the Cartesian components of the fluid velocity, R is the gas constant and T is the absolute temperature of the fluid. Therefore in the Euler limit it is enough to consider (3)

Now let w1 and w2 represent the Cartesian components of the velocity of any moving node.

In order to deal with problems involving moving nodes, and defining the derivative of F along the path of the node as

Substituting for (4)

in Eq. (3) we get

Let , be the components of the particle velocity relative to the moving node. Then Eq. (4) can be compactly written as (5)

Using MCIR [13] splitting,

and

are written as

(6)

where 1, 2 are dissipation control parameters corresponding to two components of molecular velocity. The dissipation control parameters are conveniently chosen as 1= 2=Î&#x201D;rp, where 0<p<1

@IJRTER-2016, All Rights Reserved

114

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457]

and Δr is the distance between a node and any point in its neighbourhood. Usually the closest point is chosen.

Using MCIR splitting for both the components of molecular velocity, the Boltzmann equation (5) can be written as (7)

We define moment vector function Ψ by (8)

and define the Ψ moment as (9) Now the Ψ moment of Eq. (7) will lead to the modified moving kinetic flux vector split Euler equations (10)

U is the state vector given by U=(ρ,ρu1,ρu2,ρe)T, e is the internal energy per unit mass given by and are the modified split fluxes for the moving nodes. These modified moving split fluxes are expressed in terms of moving fluxes [11] and [12] as (11)

(12)

(13)

(14)

@IJRTER-2016, All Rights Reserved

115

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457]

Further the moving split fluxes

and

can be together expressed as

(15)

where represent the split fluxes on static grid except that the velocity components in this case are relative to the node velocity and matrix [A] transforms the static fluxes to those on moving nodes.

These fluxes for finite volume formulation have been given by Krishnamurthy [8]. The transformation matrix [A] in terms of node velocities is given by (16)

The expressions for the split fluxes The x-component of is given by (17)

The y-component of

in terms of error functions and exponentials are as follows.

is given by

(18)

Here A and B are defined by

In order to develop an update scheme needed to evaluate the space derivatives of various moving split fluxes. In LSKUM_MN the space derivatives are evaluated using least squares approximation.

Consider any point Po as shown in Fig. 1. Assume that values of F are available at Po and its

@IJRTER-2016, All Rights Reserved

116

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457]

immediate surrounding nodes, referred to as connectivity of point Po. A first order approximation to the derivatives formulae

and

using weighted least squares approach [5] is then given by the following

(19)

where Δxi=xi-xo, Δyi=yi-yo, ΔFi=Fi-Fo, wi is a weight function and ∑ represents the summation over all the points in the neighborhood (i.e. connectivity) of Po. III. RESULTS AND DISCUSSIONS The aim of the present work is to compute flow past NACA0012 airfoil for prediction of lift characteristics. Lift curve slope is an important aerodynamic design input; hence accurate prediction of this is very essential. In this work we have computed the lift characteristics for two mach no’s, 0.3 and 0.35. For both cases the drag predicted is practically negligible. The experimental results are available in terms of normal co-efficient, since under these conditions the lift co-efficient is same as normal co-efficient, we have computed normal co-efficient with the experiments. All the computations have been done using a point distribution containing 9600 points; on the airfoil we have 160 points. The outer boundary is situated at approximately 10 chords distance from the surface. A set of mach no and angle of attack are then run through solver, approximately after 30,000 iterations the normal co-efficient tends to converged to a straight line. The experimental values are suggested in NASA Technical Memorandum 81927, dtd Apr 1981 by Carles D.Harris. It can be also seen that the grid free code predicts the slope of the curve very well matching with the corrected experimental results. This being an invscid code (Euler) we do not predict the stall; hence we can observe that the computations do not match at very high angle of attack, where flow separation is dominated. Apart from the normal force co-efficient, we have also compared the computed pressure distributions with the experimental values for different angles of attack. Fig (6) shows the pressure distribution comparison for different angles of attack. The angle of attack used in computation for each case is the corrected angle as suggested in reference. Again from these figures we notice that the computed distribution matches very well with the experiments. However very near the leading edge the suction peak predicted by the code doesn’t match with the experiment. This fall can be captured by using a very fine distribution of point near the leading edge.

@IJRTER-2016, All Rights Reserved

117

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457]

Lift characteristic curve for M 0.3

Pressure plot comparison for different Angles of Attack and Mach No 0.3

IV. CONCLUSIONS A lot of new developments in the grid free method have been implemented in the present work for computations of flow past bodies. Single step higher order accuracy has been achieved through the MCIR splitting and an effort is made to predicate computational values for all cases of experiments.

It is possible to reduce 2-D form to 1-D form by a suitable choice of weights. All these have been implemented in the LSKUM_MN formulation. The grid free method LSKUM_MN with all these features included has been demonstrated to work on the standard NASA test case of NACA 0012 airfoil. Further to demonstrate the power of the method, computing the unsteady flow past NACA 0012 airfoil. In the near future I hope to apply the new tool Grid free method for prediction of lift characteristics.

REFERENCES 1.

2.

V. Ramesh and S.M. Deshpande, Least squares kinetic upwind method on moving grids for unsteady Euler computations, Computational Fluids J 30 (5) (2001), pp. 621–641.

Ramesh.V “Least squares grid free kinetic upwind method”, PhD thesis, Indian Institute of science, Bangalore, July 2001.

@IJRTER-2016, All Rights Reserved

118

International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457] 3.

Deshpande SM. Kinetic theory based new upwind methods for inviscid compressible flows. AIAA Paper 86-0275, 1986.

4. Ghosh AK, Deshpande SM. Least squares kinetic upwind method for inviscid compressible flows. AIAA Paper 951735, 1995.

5. Ghosh AK. Robust least squares kinetic upwind method for inviscid compressible flows. PhD thesis. Indian Institute of Science, Bangalore, 1996.

6. Konark Arora, Deshpande SM. Weighted least squares kinetic upwind method using eigenvector basis. Fluid Mechanics Report 2004 FM 17, Centre of Excellence in Aerospace CFD. Dept. of Aero. Eng., Indian Institute of Science, Bangalore.

7. Krishnamurthy R. Kinetic flux vector splitting method on moving grids for unsteady aerodynamics and aero elasticity. PhD thesis. Indian Institute of Science, Bangalore, August 2001.

8. J.C. Mandal and S.M. Deshpande, Kinetic flux vector splitting for Euler equations, Comput Fluids 23 (1994), pp.

447â&#x20AC;&#x201C;478.

9. Ramesh V, Deshpande SM. Low dissipation grid free upwind kinetic scheme with modified CIR splitting. Fluid Mechanics Report 2004 FM 20, Centre of Excellence in Aerospace CFD. Dept. of Aero. Eng., Indian Institute of Science, Bangalore.

10. V. Ramesh and S.M. Deshpande, Least squares kinetic upwind method with modified CIR splitting, Proceedings of 7th annual CFD symposium, August 11â&#x20AC;&#x201C;12, National Aerospace Laboratories, Bangalore, India (2004).

11. Ramesh V, Mathur JS, Deshpande SM. Kinetic treatment of the far-field boundary condition. Fluid Mechanics Report 97 FM 2, Centre of Excellence in Aerospace CFD. Dept. of Aero. Eng., Indian Institute of Science, Bangalore, 1997.

@IJRTER-2016, All Rights Reserved

119