The Role of the Kutta-Joukowski Condition in the Numerical Solution of Euler Equations for a Symmetrical Airfoil.

Transcription

1 The Role of the Kutta-Joukowski Condition in the Numerical Solution of Euler Equations for a Symmetrical Airfoil. M.Z. Dauhoo Dept. of Math, University of Mauritius, Rep. of Mauritius. m.dauhoo@uom.ac.mu September, Abstract The solutions of the Euler equations are the approximate solutions of the Naviers-Stokes equations in the limit of vanishing viscosity ( viscosity µ but µ ). These solution are used to predict the lift experienced by airfoils and wings within the framework of inviscid flow, at a certain angle of attack. The classical Kutta-Joukowski hypothesis enables us to determine these solutions by imposing the Kutta-Joukowski condition at the sharp trailing edge of the airfoil. In this work, we study the question of how the circulation required for lift is produced when time marching Euler calculations are performed for an airfoil. We discuss the vorticity production, within the framework of inviscid calculation, and its role in the generation of the lift within the framework of Euler codes used in CFD. Introduction One of the fascinating aspects of aerodynamics is the classical theory of lift. According to this theory, it is possible to predict the lift of airfoils, wings and wing-bodies within the framework of inviscid theory, that is without solving the Naviers-Stokes equations of fluid flow. Sharp edges play a very crucial role in this theory of lift. For airfoils with smooth trailing edges, it is not possible to determine the lift without having recourse to the solution of the Naviers-Stokes equations. The role of the reduced Naviers-Stokes equations (with various terms neglected) in determining the approximate flow field and computing various aerodynamic coefficients cannot be ignored in design and analysis aerodynamics. It is neither always possible nor necessary to solve the Names listed in alphabetical order

2 full Naviers-Stokes equations at every stage of the design cycle and hence the importance of reduced Naviers-Stokes equations. One such approximation is the solution of the Euler equations which further reduces to the solution of Laplace s equations in the incompressible limit under the assumption that the flow is irrotational. There is one category of solutions of the Euler equations which constitutes a class of approximate solutions of the Naviers-Stokes equations in the limit of viscosity µ but µ. These solutions are termed by Lagerstrom as the relevant Euler solutions as mentioned in Salas[]. One of the main disadvantages with this type of solution is that while the prediction of lift is accurate enough, the prediction of the drag is in general subject to error. In fact, viscosity causes skin friction which always results in the production of a non negligible finite drag. The classical Kutta-Joukowski hypothesis enables us to determine the relevant Euler solution by imposing the famous Kutta-Joukowski condition, namely, the flow should leave the sharp trailing edge smoothly. This condition has been found to pick up the relevant Euler solution to a very good accuracy and has been widely used to compute the pressure distribution around airfoils, wings and wing-bodies. In the present work, for the sake of completeness, we give a brief description of the classical theory of lift and the potential flow for a circular cylinder. We consider lifting flows for an airfoil, in the framework of an inviscid compressible flow and discuss the mechanism of generating the lift for an airfoil in an inviscid fluid. We then review briefly the Euler codes, ranging from those based on the Finite Volume methods to those of node based methods, applied to lifting flows. In this review, we study how the Euler codes deal with the Kutta-Joukowski condition. We have tried to classify the different methods of implementing the Kutta-Joukowski into the Weak implementation of Kutta-Joukowski condition and Strong implementation of the Kutta-Joukowski condition []. We describe the new theory on generation of circulation [, ], when time marching Euler calculations are performed with the Kutta-Joukowski condition imposed at the trailing edge. We then discuss the vorticity production and distribution in the computational domain when an airfoil with a sharp trailing edge is immersed in an inviscid compressible flow. We finally present the results of our numerical experiments. Euler codes applied to lifting flows During the early development of Euler codes in CFD, it was found that the explicit imposition of the Kutta-Joukowski condition was not necessary, that is, no special care was required to ensure that the flow leaves the trailing edge smoothly. Many of those codes were cell centered finite volume codes which update the cell averages at the centroids of cells or finite volumes. The centroids of the cells in this formulation do not lie on the solid boundary and thus, the need to update the state variables exactly at the sharp trailing edge is avoided. In a way, the Euler codes based on cell-centered finite volume method avoid the sharp trailing edge (to be called the Kutta point hereafter). The problem of how to obtain the flow variables at the Kutta point is not addressed

3 in this formulation. Even then, these codes have been used and are being widely used to compute the flow field and obtain the aerodynamic coefficients quite accurately. A fluid dynamically interesting question arise at this point regarding how the Kutta-Joukowski condition is satisfied in this formulation: Is it unnecessary because it is automatically satisfied? Let us look at some of the issues involved in more details. We consider the case of a -D airfoil with a sharp trailing edge and further consider finite volumes near the trailing edge and elsewhere on the body (Fig.(8)). Fig.(8) shows five cells with centroids a, b, c, d and e. While updating the state variables at the centroid a, it is necessary to compute the fluxes on all the edges or faces of the cell. In particular, it is required to determine the flux on the cell face lying on the body (shown hatched in Fig.(8)). On this face, flow tangency boundary condition has to be imposed and this can be done in many ways. One way is to use the Kinetic Characteristic Boundary Condition based on the reflection principle [7]. This cell face does not pose any special problem in implementing the solid wall boundary condition. Now consider the cell with centroid b. One of its edges is (, ) and the node is the Kutta point. While imposing the solid wall boundary condition on that edge, a problem arises because all along (, ), the flow tangency boundary condition has to be imposed. But at the Kutta point (node ), we have two flow tangency boundary conditions - one from above corresponding to the normal vector on the upper surface(n u )and one from below corresponding to the normal vector on the lower surface(n l ). For bodies with a finite angle at the Kutta point, these two normals are different and if q te is the velocity vector at the Kutta point then, we get q te n u = q te n l = () implying that q te =, that is, u = u =, the velocity components at the Kutta point are zero. In this argument, it is assumed that the fluid velocity is single valued. For subsonic flows, q te = in the steady state. For transonic flows with a shock present on the lower or upper surface, this assumption is not valid. We shall study the condition at the trailing edge in transonic flow later. In all finite volume codes that we know of, the velocity vector is not set to zero at the Kutta point. How is the flow tangency boundary condition implemented is then the root question. In first order accurate computations, the flow variables are assumed to be constant within a cell. A part of the flux on the cell face (, ) coming from the interior (that is, inside the computational domain), is taken to be the same as that at the centroid b; and the other part of the flux coming from the exterior (outside the computational domain), is obtained from the Kinetic Characteristic Boundary Condition or the reflection principle. Incodesbasedonlinear reconstruction (required for enhancing order of accuracy) the flow variables are assumed to vary linearly within a cell, and this can be done by obtaining the gradient of the flow variables at centroid b, that is, for any quantity f(x, y) we assume ( ) ( ) f f f (x, y) =f (x b,y b )+(x x b ) +(y y b ) () x b y b

4 ( ) ( ) f f The gradient terms and are determined from the neighbourhood data and again, no attention is given to the Kutta point. The main point x b y b is that the solid boundary condition suddenly changes from q te n u = (at a point vanishingly close to the trailing edge) to q te n l = at a point on the lower surface but vanishingly close to the Kutta point. Many schemes of implementing the solid boundary condition do not take into account this sudden change in boundary condition. More or less the same criticism is valid for all cell centered finite volume codes. It is interesting and important to note that these Euler codes give satisfactory results in spite of the above criticism. How then does the fluid leave the trailing edge smoothly in these computations? It is argued that the centroids c and d communicate with each other via the cell face (, ). The flux on the cell face (, ) is the same in the state update for centroids c and d. So, the argument can be expressed as follows : The flow will be prevented from suddenly turning around the Kutta point thus leading to the satisfaction of the Kutta-Joukowski condition. It is evident that the question of determining the values of pressure and density at the Kutta point is left open in these formulations. One can also determine these values from the neighbouring data but it has been found that Euler codes give widely different values of the coefficient of pressure at the trailing edge (C pte ). In the cell vertex schemes based on dual finite volume approach (sometimes called covolume methods), it is not possible to skirt the issue of applying the state update at the Kutta point. A typical dual finite volume around the Kutta point is shown in Fig. (8). Again the problem of determining the fluxes on the cell faces lying on the solid wall arises. The necessity of updating the state variables at the Kutta point comes into sharper focus for the LSKUM [7] and the q-lskum []in which we obtain the space derivatives at the Kutta point (Fig. () in terms of the neighbouring data. The basic question is : Given the flow variables ρ n kp, un kp, v kp n, pn kp at time level n at the Kutta point, how to determine ρ n+ kp, u n+ kp, v n+ kp, p n+ kp using the data in the connectivity of the Kutta point. In order to solve this problem, we need to find the solution of the partial differential equation of the flow together with the prevailing boundary condition. It is thus essential to have a clear mathematical way for updating the state at a sharp edge singularity. This formula will obviously be somewhat different from that at an interior non-singular point. We now introduce the following definitions : (i)weak implementation of the boundary condition at the Kutta point (ii)strong implementation of the boundary condition at the Kutta point The Weak implementation is employed in the Cell Centered Finite Volume Method, where the question on state update at the Kutta point, has been discussed above. Mathur [9] and many others have extensively employed the Weak implementation of the Kutta condition in all their calculations. In the strong implementation, the state update is directly applied to the Kutta point. A question can now be raised regarding airfoils with a cusp at the trailing edge

5 instead of a finite angle at the trailing edge considered so far. In such a case, n u = n l () and q te n u = implies q te n l =. Hence, we can not conclude that q te = at a cusp, even under the assumption of a single valued fluid velocity. Suppose the angle at the trailing edge is θ then we have the following mathematical situation q te (θ) = for θ > () q te (θ) n u = for θ = () Assuming single valued q, wehaveq te (θ) = however small θ is and we do not know whether it is zero for θ =. If we assume that q te is a continuous function of θ, then, we can assert that the condition () is still valid and does not contradict condition (). The validity of condition () for θ = hasto be investigated and till then, no definite conclusion can be drawn. All the calculations made in this work are for θ, that is, the airfoils have a finite angle at the Kutta point. In this section, we are going to examine closely the flow variables at the trailing edge. We shall be using the following notations: P ote : total pressure at the Kutta point. p te : static pressure at the Kutta point. P ol : total pressure at a point L on the lower surface and vanishingly close to the trailing edge. P ou : total pressure at a point U on the upper surface and vanishingly close to the trailing edge. p te,l : static pressure at the L. p te,u : static pressure at the U. M te,l : Mach number at L. M te,u : Mach number at U. q te,u : The fluid velocity at U. q te,l : The fluid velocity at L. By the principle of continuity of static pressure near the trailing edge, we have p te,l = p te,u = p te () From this common value of static pressure at the trailing edge, we can obtain P ol and P ou by using the isentropic relations and P ol = p te,l ( + P ou = p te,l ( + ) γ (γ ) Mte,L γ (7) ) γ (γ ) Mte,L γ. (8) Evidently, we need to know M tel and M teu in order to get P L and P U. We now consider the subsonic case and transonic case separately since for the former, the fluid variables are single-valued while for the latter, some of them are not.

6 . Subsonic case Since streamlines cannot suddenly turn at the Kutta point in a subsonic flow, and further, assuming single-valued property of the flow variables, it is obvious that the Kutta point is a stagnation point. Thus, q te = and there is no loss in total pressure as we move along a streamline. Hence, Eq.(8) is satisfied because M te =.. Transonic case p te = P ol = P ou = P o (9) Let us now assume that a transonic flow is prevailing at steady state and in which there is a shock on the upper surface of the airfoil as shown in Fig.(??). Because of the shock on the upper surface, P ou < P o and assuming no shock on the lower surface, we obtain P ol = P o. By Eq.(), P ou p te < P ol p te () and using Eq(7) and Eq.(8), M te,u M te,l. () Since the streamlines can not suddenly turn at the Kutta point, we can assume as before that M teu = M tel =. This contradicts Eq.(). We therefore give up the assumption that M teu = M tel =. We consider another possibility, that is, the stagnation point is located at L and the flow is smooth at U (see Fig.()). We have P ou < P ol = P o, that is, P ou < () P ol However, Eq.() contradicts the fact that the static pressure is always less than the stagnation pressure at any point within the computational domain, particularly at L. The only possibility left is the stagnation point located at U and we shall show that it is indeed the case that the fluid stagnates when the shock is found on the upper surface of the airfoil. Since U is a stagnation point, it follows that p te,u = P o,u. () Using Eq.() and Eq.(), P ou = p te and P ol = P ( ol = + P ote P ou ) γ (γ ) Mte,L γ > () As a result, q te,l and this is consistent with the last assumption. We can apply the same reasoning in a transonic case with a shock on the lower surface and show that L is a stagnation point. We can therefore summarize the flow near the trailing edge as follows :

7 In subsonic flow, the Kutta point is a stagnation point. For all the flow situations considered, the flow leaves the trailing edge smoothly thus satisfying the Kutta-Joukowski condition. For the transonic case, except from the pressure, all the other flow variables are discontinuous across the thick horizontal line shown in Fig.(8). This suggests that continuity of pressure is one of the criterion that can be enforced in order to satisfy the Kutta-Joukowski condition. We shall next analyze different ways of enforcing the Kutta-Joukowski condition when using a node based method while taking into account the state of the flow at the trailing edge as just described. Strong implementation of the Kutta-Joukowski condition In a transonic flow with a shock on the upper surface, the stagnation point is found at U and not at L. As mentioned before, the Cell Centered Finite Volume method based Euler codes do not take any special care to enforce the Kutta- Joukowski condition and we have termed this as the Weak implementation of the Kutta-Joukowski condition. The node based schemes such as LSKUM, q-lskum and rotated q-lskum require the Strong implementation of the Kutta-Joukowski condition. In this paper, we have made an attempt to address the question of state update at the Kutta point directly. The q-lskum [] based Euler code has been used to compute subsonic flows around the NACA airfoil. The boundary condition: q te = (θ ), at the Kutta point has been implemented while taking into account the continuity of static pressure at that point and the fact that the flow should leave the trailing edge smoothly: ThePressure Least Squares Interpolation Method (PLSI ) []. In the PLSI method, the state variables ρ n+ kp, u,kp n+, u n+,kp and e n+ kp at the Kutta point, are updated first as an interior node and then the pressure is overwritten using interpolation, that is, p n+ kp is obtained from least squares interpolation using the neighbouring data. The Kutta-Joukowski theorem and the generation of lift When the method of conformal transformation is used in order to obtain lifting flows around airfoils, it is essential to fix Γ by some criterion. The Kutta- Joukowski condition is known to fix the circulation (hence the lift) and to yield the relevant Euler solution as per the classical theory of lift[8]. It has been found that when the rear stagnation point is fixed at the trailing edge, the flow leaves the trailing edge smoothly. The Kutta-Joukowski condition can therefore be considered as that condition which gives a smooth flow near the trailing edge[?]. We want to study how modern Euler codes applied to the computations of the compressible inviscid flows generate vorticity and produce enough circulation, even though we do not add explicitly any circulation to the 7

8 computed flow. Vorticity production due to the baroclinic mechanism Let us first describe briefly how vorticity is produced in viscous flows. We consider a viscous flow over a flat plate (Fig. ), the no-slip condition is applied on the flat plate. Thus, there is a region of thickness, say δ, over which the viscous effects are important and this is due to the fact that the fluid is continuously brought to rest on the plate. Hence, the velocity changes sharply from its no slip value of zero at the solid surface to a non zero value at the edge of this layer(fig.()). The small thickness of this layer ensures a large velocity gradient, u, in the normal direction. Vorticity is therefore produced near the y flat plate and once produced, it is convected, stretched (for D) and diffused. In the case of Euler codes used for computing lifting flows around airfoils, we do not add any circulation as in the classical theory of lift and the above-mentioned no-slip boundary condition is not present in order to produce vorticity by the action of viscosity. A question then arises at this point : How is the circulation necessary for lift generated? According to a recent study by Balasubramaniam et al [], the circulation necessary for lift is produced by the baroclinic mechanism. The modern Euler codes, for computing lifting flows, reach the steady state via time marching. According to the above proposed theory, the baroclinic mechanism is turned on during time marching and produces vorticity and therefore circulation. In the numerical computation of a subsonic flow field for an inviscid fluid past an airfoil with a sharp trailing edge, the Kutta-Joukowski condition can be regarded as a kind of one point no-slip boundary condition, that is, q t.e. = (for θ = ) at the Kutta point.we now analyse the relationship between the production of vorticity in the subsonic flow for an inviscid fluid past an airfoil with a sharp trailing edge and the boundary condition at the latter point. Balasubramaniam et al [] has proposed that the sudden change in boundary condition causes vorticity production through the baroclinic mechanism. In order to understand the baroclinic mechanism, we must refer to the vorticity transport equation. We consider the momentum equation for an inviscid compressible flow. It is given by q + q. q = p () t ρ where q is the fluid velocity, ρ is the density and p the pressure. Taking the curl of both sides of Eq.(), we get ω ( ) ( ) + q. q = t ρ p () For a -dimensional case, the above Eq. reduces to ω ω + u t x + u ω y = [ ] ρ p (ρ) (7) 8

9 The left hand side of Eq.() has the time rate of change of ω and the term giving vorticity advection. The right hand side of Eq.() vanishes when p and ρ are isentropically related. If p is not parallel to ρ, then this term produces vorticity. Such a production of vorticity takes place through what is termed as the baroclinic mechanism. For inviscid flows with the Kutta-Joukowski condition prevailing at the trailing edge, there is a sudden discontinuity in the wall boundary condition. At the sharp trailing edge, the Kutta-Joukowski condition requires u = and u =, that is, the velocity vector is zero at the trailing edge while flow tangency boundary condition at the other nodes on the body requires only the normal component of velocity to be non-zero (q n = q n = ). As a result, sharp velocity gradients are developed. A non-isentropic change thereby takes place in the flow field because the fluid elements in the neighbourhood of the Kutta-point are brought to rest abruptly at the trailing edge. Thus, strong gradients in pressure and density are set up and they in turn produce vorticity. Another example where vorticity is produced via the baroclinic mechanism is in the case of a supersonic flow(m > ). As shown in Fig.(8), there is a Bow Shock ahead of the [ leading edge ] and the flow is no more isentropic in region R. Therefore, p (ρ) and hence, vorticity is produced in that region. Our numerical experiments show that the term ] [ p ρ (ρ) is very large near the trailing edge. As mentioned earlier in this section, it is believed that the sharp trailing edge triggers this term and vorticity production takes place at the Kutta point. In these experiments [], we note that the vorticity is a maximum at the trailing edge. As stated earlier, the lift experienced by an airfoil in a subsonic flow at a given angle of attack is a consequence of the circulation around the airfoil. Consequently, the existence of the circulation is given by Γ = q.ds = ω.ds (8) c s where ω = q, q being the velocity vector at the corresponding node, implies that the vorticity ω must be non-zero in the domain. In the numerical experiments performed using q-lskum based Euler code and PLSI at the Kutta point, the vorticity at each node of the domain is computed when the steady state is reached. It is observed that the vorticity is a maximum at the trailing edge when compared to the rest of the domain. The numerical value of the vorticity at the trailing edge increases sharply as the grid is refined.even though the vorticity is everywhere zero (except at one point) we still get a finite circulation around the airfoil. For low Mach number computations also, that is, in the incompressible limit, our numerical investigations show that the baroclinic mechanism is present (Fig.(??) and Fig()). In other words, the flow is always compressible in the neighbourhood of the trailing edge. The baroclinic mechanism requires compressibility and it is interesting to note its presence near the sharp edge even near the incompressible limit. 9

10 7 The baroclinic term Fig.() shows the distribution of the baroclinic term for x (coarse) and x9 (medium) grids respectively in steady state. It is clear that the baroclinic term, is numerically highest at the leading edge; and it is quite large at the trailing edge. The latter observation confirms that after the vorticity is being produced (due the baroclinic term (Eq. (7), it is convected along the airfoil and it accumulates at the trailing edge where it is found to be numerically largest. Our computations begin with impulsively started initial conditions, that is, velocity is set to free stream velocity everywhere in the domain at t = and then at t = + its normal component on the surface of the airfoil suddenly becomes zero. Very large gradients of velocity, density and pressure are thereby generated close to the surface of the airfoil, causing large values of baroclinic term and these in turn produce vorticity on the surface of the airfoil. Fig.() shows that the baroclinic mechanism is active on the surface of the airfoil even in steady state. The same trend is confirmed at low Mach number, the baroclinic term is large in the suburb of the solid boundary (Fig.( )). Also, In particular, it is large near the leading edge and the trailing edge. Also, Fig. show that the divergence is very large at the trailing edge even in the incompressible limit. It is in fact a maximum at the trailing edge, confirming that sharp velocity gradients are developed in the region close to the Kutta point even at low Mach number. 7. Vorticity plots at steady state for different grid sizes In order to study the vorticity distribution at steady state, we have chosen a subsonic flow around NACA airfoil with M =.,α =. The two dimensional distributions of vorticity at steady state, for relatively coarse (x), medium (x9) and fine (x) grid respectively, are shown in Fig.(). Also, we computed the flow field for a fine grid of 88 nodes (x) atlowmachnumberthatis,m =.,α= o. Our results suggest that as the grid is progressively refined, the vorticity will tend to a Dirac function, that is, it will be very large at the Kutta point and almost zero everywhere in the domain. We believe that vorticity is generated along the airfoil and is then convected to the trailing edge (q t.e. = ) where it accumulates in the steady state. Also, from Fig.(7) and Fig.(8)), it can be seen that the vorticity is highest on the surface of the airfoil at steady state. Thus, the solution of the compressible Euler equations reduces to that of the potential equation in the steady state. That is, the surface of the airfoil can be approximated by a layer of vortices. 8 Conclusions We have made an attempt to understand how the circulation necessary for lift is generated within the framework of inviscid compressible flow. It is suggested that the baroclinic mechanism, activated by flow tangency and the Kutta-Joukowski condition, is responsible for generating vorticity and therefore

11 circulation. Our grid refinement study shows that the vorticity distribution at steady state increases at the trailing edge. That is, vorticity is either produced by the baroclinic mechanism or near the solid boundary by viscosity. We conclude by proposing the following tentative hypotheses []: Hypothesis () : For inviscid compressible flows past lifting airfoils, vorticity is produced by the baroclinic mechanism. Hypothesis () : For inviscid compressible flows past lifting airfoils, the vorticity is very large at the trailing edge. Acknowledgements The author is thankful to the University of Mauritius, Rep. of Mauritius and the CFD Centre, IISc, Bangalore,India for this work. References [] Salas M.D., Foundations for the Numerical solutions of the Euler equations, Recent advances in Numerical Methods in Fluids, W.G. Habashi, Editor, Pinewood Press Ltd., Swansea, U.K., pp. 87-9, 98. [] Balasubramaniam R., Numerical experiments with Higher Order Least Squares Kinetic Upwind Method, M.E. Thesis, Indian Institute of Science, Bangalore, India, 997. [] Balasubramaniam R., Ramesh V. and Deshpande S.M., On Kutta-Joukowski Condition, The Seventh Asian Congress of Fluid Mechanics, Chennai, India, December 997. [] Dauhoo M.Z., Raghuramarao S.V., Ramesh V. and Deshpande S.M., A Strong Implementation of the Kutta-Joukowski Condition using Peculiar Velocity Based Upwind Method, th ICNMFD,Lecture Notes in Physics, Springer, 998. [] Dauhoo M.Z., Ghosh A.K., Ramesh V. and Deshpande S.M., q-lskum a new Higher Order Kinetic Methods For Euler Equations Using The Entropy Variables, 8th International Symposium on Computational Fluid Dynamics, Bremen(Germany), September [] Dauhoo M.Z.The Least Squares Kinetic Upwind Method based on Entropy variables and its Application to the Understanding of the Aerodynamics of Lift, Ph.D. Thesis, Dept. of Mathematics, Faculty of Science, University of Mauritius, July. [7] Ghosh A.K. and Deshpande S.M., A Robust Least Squares Kinetic Upwind Method for Inviscid Compressible Flows, AIAA Paper No [8] John Andersson, Jr., Modern Compressible Flow With Historical Perspective, Mc Grawhill International Editions, Aerospace Series, 99. [9] Mathur J.S. and Deshpande S.M., Reconstruction on unstructured grids using an upwind kinetic method, Proc. th Int. Conf. on Numerical Methods in Fluid Dynamics, Monterey, California, June -8, 99. [] Milne-Thompson L.M., Theoretical Aerodynamics, Third Edition, London, Macmillan and Co. LTD., New York, St Martin s Press, 98.

12 [] Raghurama Rao S.V., New Upwind Methods Based on Kinetic Theory for Inviscid Compressible Flows, Ph.D. thesis, Indian Institute of Science, Bangalore, India, 99. a θ b c e d 7 (a) The Kutta-Joukowski condition in the Finite Volume Methods (b) Dual cell around the Kutta point. U U L L (c) The Kutta point considered as a stagnation point in the transonic flow (d) L considered as a stagnation point in the transonic flow Figure : Shock U. α 8U L Region R M > ω (b) Stagnation point at U in transonic flow Bow Shock (a) Vorticity Production by the Baroclinic mechanism due to non-isentropic flow

13 KP Y Boundary Layer U X 8 (c) The connectivity at the Kutta point Flat Plate (d) Viscous Flow over a Flat Plate Figure : Figure : Surface Plots of the Baroclinic term, at steady state, a coarse and a medium grid respectively ( M =., α = ) Figure : Surface Plots of Vorticity, at steady state, for different grids. ( M =., α = )

14 (a) The whole Computational Domain (b) Close to the Airfoil (c) The Leading edge (d) The Trailing edge Figure : The Baroclinic term for the flow past NACA at M =., α =. o on Fine Grid (a) The whole Computational Domain (b) Close to the Airfoil

15 (c) The Leading edge (d) The Trailing edge Figure : Contours of Divergence for the flow past NACA at M =., α =. o on Fine Grid 8 x Figure 7: A -D Surface plot of the Vorticity distribution close to the airfoil for the coarse grid x 8 x Figure 8: A -D Surface plot of the Vorticity distribution (fine grid) close to the airfoil in the limit of incompressibility for the flow past NACA at M =., α =. o