THE INITIAL FLOW PAST A UNIFORMLY ACCELERATING INCLINED ELLIPTIC CYLINDER

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 8, Number 3, Fall 2 THE INITIAL FLOW PAST A UNIFORMLY ACCELERATING INCLINED ELLIPTIC CYLINDER S.J.D. D ALESSIO AND F.W. CHAPMAN ABSTRACT. This paper solves the problem of determining the initial two-dimensional motion of a viscous incompressible fluid past an inclined elliptic cylinder which is uniformly accelerated from rest. This motion is calculated using two types of methods. The first takes the form of a double series solution where an expansion is carried out in powers of the time, t, andinpowersofλ = 8t/R where R is the Reynolds number. This approximate analytical solution is valid for small times following the start of the motion and for large Reynolds numbers. The second method involves a spectral-finite difference procedure for numerically integrating the full Navier-Stokes equations expressed in terms of a stream function and vorticity. Our results demonstrate that for small times and moderately large R the two methods of solution are in good agreement. Results are presented for the cases R = 1 and R =1, for an ellipse inclined at 45 to the free stream and having an aspect ratio of Introduction. This paper discusses the two-dimensional problem of the flow of a viscous incompressible fluid past a uniformly accelerating cylinder. The cylinder is taken to be elliptic in cross section and inclined at an angle of α with the horizontal. The cylinder starts from rest and then accelerates uniformly in the horizontal direction. A similar study conducted by Collins and Dennis [1]considered the simpler case of symmetrical flow past a uniformly accelerated circular cylinder while Badr, Dennis and Kocabiyik [2]more recently investigated the case where the cylinder speed varies according to U(t) =U + U 1 t + U 2 t 2. Although much has been written about the circular cylinder, relatively little work has been devoted to the case of the elliptic cylinder. This is probably due to the more complicated geometry as well as the fact that the flow is asymmetrical and therefore intrinsically more complex. A comprehensive list of references detailing the experimental, numerical and theoretical properties of such flows can be found in Badr and Dennis [3]. The unsteady development of flow past a uniformly accelerating Copyright c 2 Rocky Mountain Mathematics Consortium 215

2 216 S.J.D. D ALESSIO AND F.W. CHAPMAN elliptic cylinder is of interest for both theoretical and practical reasons since it can be related to engineering applications such as the start-up problem associated with the flow past an airfoil. Understanding the flow and aerodynamic forces experienced by the airfoil will be of use to the aeronautical industry. The flow variables chosen to describe the motion are taken to be the scalar stream function and vorticity. The main purpose of the present study is to reconcile numerical solutions of the Navier-Stokes equations with an analytic solution which is valid for small times following the start of the motion and for large Reynolds numbers, R. The numerical technique employed is a spectral finite difference scheme and is successful in capturing the early development of the flow. The paper is organized as follows. We begin by presenting the governing equations along with their corresponding initial and boundary conditions and introduce a coordinate system which is best suited for the geometry at hand. Then, we rescale the problem with a choice of variables which are suggested by boundary-layer theory and perform an expansion in powers of the time, t, and in terms of the parameter λ = 8t/R. This procedure yields a series solution for the flow variables which is valid for small time departures from the start of the motion and for large values of R. Following this, a numerical method for solving the governing equations is provided. Lastly, results are presented whereby the numerical integrations are contrasted against the series expansion for derived quantities such as drag, lift and pressure coefficients. 2. Governing equations and boundary conditions. We will solve the equivalent problem of a stationary inclined elliptic cylinder immersed in a fluid that is uniformly accelerating past it. This simplifies the mathematical formulation of the problem. The elliptic cylinder whose axis coincides with the z-axis is fixed at the origin and is inclined at an angle α with the horizontal. At t = the free stream is allowed to uniformly accelerate at a rate b horizontally from right to left. For two-dimensional flow the governing Navier-Stokes equations can be expressed in terms of a stream function ψ and scalar vorticity ζ. The dimensionless functions ψ and ζ are related to their dimensional

3 INITIAL FLOW PAST A CYLINDER 217 counterparts ψ and ζ through (1) ψ = c cbψ, ζ = cbζ/c where 2c is the focal length of the ellipse. In terms of the dimensionless velocity components u, v, obtained by dividing the corresponding dimensional components by cb, wehavethat (2) u = ψ y, v = ψ x, ζ = u y + v x. Thus, the equations of motion may be expressed by two simultaneous, coupled, nonlinear partial differential equations for ψ and ζ. In Cartesian coordinates these equations assume the form (3) 2 ψ x ψ y 2 = ζ (4) ζ t = ψ ζ x y + ψ y ζ x + 2 ( 2 ) ζ R x ζ y 2. The dimensionless time t is related to the dimensional time t through t = bt / c and the Reynolds number R is defined as R =2c cb/ν, with ν being the kinematic viscosity. Because the Cartesian coordinate system is not convenient for numerical work, we introduce the conformal transformation (5) x + iy =cosh[(ξ + ξ )+iθ] which transforms the contour of the cylinder to ξ = and the infinite region exterior to the cylinder to the semi-infinite rectangular strip ξ, θ 2π. The constant ξ appearing in (5) is defined by (6) tanh ξ = r where r is the aspect ratio specifying the geometry, i.e., the ratio of the minor to major axis of the ellipse. The x and y axis are oriented along

4 218 S.J.D. D ALESSIO AND F.W. CHAPMAN the major and minor axis of the ellipse respectively. In the transformed coordinates (ξ,θ) equations (3) and (4) become (7) (8) 2 ψ ξ ψ θ 2 = M 2 ζ ζ t = 1 [ M 2 ψ ζ ξ θ + ψ ζ θ ξ + 2 R ( 2 )] ζ ξ ζ θ 2, where M refers to the metric of the transformation, given by (9) M 2 = 1 2 [cosh 2(ξ + ξ ) cos(2θ)]. The velocity components (v ξ,v θ ) in the directions of increase of (ξ,θ) now become (1) v ξ = 1 M ψ θ, v θ = 1 M ψ ξ, and the vorticity is found through (11) ζ = 1 M 2 ( θ (Mv ξ)+ ξ (Mv θ) ). Boundary conditions for ψ and ζ include the no-slip condition (12) ψ =, ψ ξ = on ξ =, which states that the fluid must come to rest on the cylinder surface, and the periodicity condition (13) ψ(ξ,θ,t) =ψ(ξ,θ +2π, t), ζ(ξ,θ,t) =ζ(ξ,θ +2π, t), which states that the flow variables are periodic with respect to θ. At large distances we demand that (14) ξ ψ e ξ 1 2 teξ sin(θ + α), e ξ ψ θ 1 2 teξ cos(θ + α), ζ as ξ.

5 INITIAL FLOW PAST A CYLINDER 219 Condition (14) is referred to as the far-field condition, which states that far away from the cylinder the fluid velocity increases linearly with time in accordance with the uniform rate of acceleration. By inspecting the boundary conditions we see that there are two conditions for the stream function on the cylinder surface while none for the vorticity. A method of determining the surface vorticity involves the use of integral conditions, which we now introduce. We begin by recalling Green s second identity, which states that any differentiable functions g and h throughout the region D must obey (15) (g 2 h h 2 g) dx dy = D C ( g h n h g ) ds. n Here, h/ n, g/ n refer to normal derivatives and C is the closed curve corresponding to the boundary of D. Choosing g = ψ and h to be harmonic functions given by (16) h n = e nξ sin(nθ) and h n = e nξ cos(nθ), and making use of condition (14), Green s second identity becomes (17) 1 2π e nξ M 2 ζ sin(nθ) dθ dξ = te ξ cos(α)δ 1,n, n =1, 2, π 1 2π e nξ M 2 ζ cos(nθ) dθ dξ = te ξ sin(α)δ 1,n, n =, 1, π where δ 1,n is the Kronecker delta, defined by { 1 if n =1 (18) δ 1,n = if n 1. These conditions are of a global or integral nature. Basically, Green s second identity enables us to convert the boundary conditions on the surface and at large distances into conditions that are valid throughout the entire domain. Further use of Green s theorems in fluid mechanics is discussed in Dennis and Quartapelle [4]. Later, we will explain how these integral conditions can be used in determining the surface vorticity. 3. Determination of the initial flow. The initial flow is governed by the usual boundary-layer theory proposed by Blasius [5],in which

6 22 S.J.D. D ALESSIO AND F.W. CHAPMAN a thin layer of thickness 2 2t/R surrounds the cylinder. As a result of this, we make the following boundary-layer type transformation in order to deal with the initial flow: 8t (19) ξ = λz, ψ = λψ, ζ = ω/λ, λ = R. In this transformation, λ denotes the growth of the boundary-layer structure of the initial solution. This change of variables stretches the thin boundary-layer and scales the stream function and vorticity so that they are of order unity. This transformation has been successfully used in previous studies including Collins and Dennis [6, 7],Staniforth [8], Dennis and Staniforth [9], Badr and Dennis [3], and more recently by D Alessio, Dennis and Nguyen [1]. In terms of these new variables, equations (7) and (8) now become (2) 2 Ψ z 2 + λ2 2 Ψ θ 2 = M 2 ω (21) 1 2 ω ω M 2 +2z z2 z +2ω =4t ω t λ2 M 2 2 ω θ 2 4t M 2 ( Ψ θ ω z Ψ z ) ω, θ and will be used to dictate the early stages of the flow. We emphasize that although boundary-layer coordinates are utilized, the full Navier-Stokes equations are to be solved and not merely the simplified boundary-layer equations. If R is large and t is small, then λ is also small, and it is possible to expand the flow variables in a power series in t. The expansions for Ψ and ω canbemadeintermsofbothλ and t. Firstly, we expand Ψ and ω in a series of the form (22) Ψ=Ψ + λψ 1 + λ 2 Ψ 2 + ω = ω + λω 1 + λ 2 ω 2 +. Then each Ψ n, ω n, n =, 1, 2,, can be further expanded in a series of the form (23) Ψ n (z, θ, t) =Ψ n (z, θ)+tψ n1 (z, θ)+ ω n (z, θ, t) =ω n (z, θ)+tω n1 (z, θ)+.

7 INITIAL FLOW PAST A CYLINDER 221 Initially, when t<8/r, wehavethatλ>t. However, once t>8/r, we have that t > λ. The parameters λ and t are equal when the dimensionless time t = 8/R. The first terms in the series (22) Ψ and ω can be obtained by setting λ = in equations (2) (21). This leads to the boundary-layer equations, since they result in the limit as R and are valid for all time. These equations are (24) 2 Ψ z 2 = M 2 ω (25) 1 M 2 2 ω z 2 +2z ω z +2ω =4t ω t 4t M 2 ( Ψ θ ω z Ψ z ) ω, θ where (26) M 2 = 1 2 [cosh(2ξ ) cos(2θ)]. The initial solution to equation (25) at t =, denoted by ω,satisfies the condition that ω asz, as well as the integral conditions (27) 1 π 1 π 2π 2π M 2 ω sin(nθ) dθ dz =, n =1, 2,... M 2 ω cos(nθ) dθ dz =, n =, 1, 2,. This yields the trivial solution ω =. With this, the initial solution Ψ to the stream function satisfies (28) subject to 2 Ψ z 2 = (29) Ψ = and Ψ z = atz =,

8 222 S.J.D. D ALESSIO AND F.W. CHAPMAN and also yields the trivial solution Ψ =. Thus, the non-zero leading-order terms for Ψ and ω are given by Ψ 1 and ω 1 and are governed by the equations (3) (31) 1 M 2 2 Ψ 1 z 2 = M 2 ω 1 2 ω 1 z 2 +2z ω 1 z 2ω 1 =. In order to determine the integral conditions which ω 1 must obey, we substitute the Taylor series expansions exp( nλz) =1 nλz + and M 2 = M 2 +sinh(2ξ )λz +... along with expansion (22) for the vorticity into (17). Equating all terms that are O(t) leadsto 1 π 2π M 2 ω 1 sin(nθ) dθ dz =exp(ξ )cosαδ 1,n, n =1, 2,... (32) 1 π 2π M 2 ω 1 cos(nθ) dθ dz =exp(ξ )sinαδ 1,n, n =, 1, 2,. The solution satisfying the above integral conditions and the far-field condition ω 1 asz is (33) ω 1 (z, θ) = 4eξ where M [ ] e M 2 z2 M z erfc (M z) sin(θ + α), π (34) erfc (x) =1 erf (x) and erf(x) = 2 x e u2 du. π After some algebra, the corresponding solution to the stream function subject to (35) Ψ 1 = and Ψ 1 z = atz =

9 INITIAL FLOW PAST A CYLINDER 223 becomes (36) Ψ 1 (z, θ) = eξ M [ M z erf (M z) 2 3 M 3 z 3 erfc (M z) π 3 π M 2 z 2 e M 2 ] 2 (1 e M z2 ) sin(θ + α). The transverse component of velocity v θ is given by (37) v θ = 1 ψ M ξ. Using the approximation Ψ tψ 1 yields the result (38) v θ = 2teξ M [ 1 2 erf (M z)+ M ] z e M 2 z2 + M 2 π z2 erfc (M z) sin(θ + α). In the limit as z we obtain (39) v θ = teξ M sin(θ + α), which is consistent with the transverse velocity predicted by potential flow evaluated at the cylinder surface. The full potential flow solution is presented in the Appendix. Hence, if R is large enough and t small, the solution is given approximately by z2 (4) Ψ tψ 1 ω tω 1. The above boundary-layer result is limited to both small time and infinite R. In order to account for large but finite values of the Reynolds number, higher-order corrections to Ψ and ω are necessary. However, the process of deriving analytical expressions soon becomes quite complicated as we will demonstrate. The first-order correction comes from Ψ 1 and ω 1 and satisfies (41) 2 Ψ 1 z 2 = M 2 ω 1 +sinh(2ξ )zω

10 224 S.J.D. D ALESSIO AND F.W. CHAPMAN (42) 2 ω 1 z 2 +2M 2 z ω 1 z = 2sinh(2ξ )z 2 ω z 2sinh(2ξ )zω +4M 2 t ω 1 t +4sinh(2ξ )zt ω ( t Ψ1 4t θ Ψ z ω z + Ψ θ ω 1 z ω 1 θ Ψ 1 z ) ω. θ Without much difficulty, we showed that the leading order terms in the expansions for Ψ 1 and ω 1 satisfying all the necessary conditions are the trivial solutions Ψ 1 =andω 1 =. Thus, the perturbation solution given by (4) can be built up as (43) Ψ tψ 1 + λtψ 11 ω tω 1 + λtω 11, or more precisely we can write Ψ=tΨ 1 + λtψ 11 + O(t 2 + λt 2 + λ 2 ) ω = tω 1 + λtω 11 + O(t 2 + λt 2 + λ 2 ). The functions Ψ 11 and ω 11 are determined from (44) 2 Ψ 11 z 2 = M 2 ω 11 +sinh(2ξ )zω 1 (45) 2 ω 11 z 2 +2M 2 z ω 11 z 4M 2 ω 11 = 2sinh(2ξ )z 2 ω 1 z +2sinh(2ξ )zω 1, and must satisfy the boundary and far-field conditions (46) Ψ 11 = Ψ 11 z =atz = andω 11 as z,

11 INITIAL FLOW PAST A CYLINDER 225 as well as the integral conditions 2π = M 2 ω 11 sin(nθ) dθ dz 2π [nm 2 sinh(2ξ )]zω 1 sin(nθ) dθ dz, n =1, 2,... (47) = 2π 2π M 2 ω 11 cos(nθ) dθ dz [nm 2 sinh(2ξ )]zω 1 cos(nθ) dθ dz, n =, 1, 2,.... The solution for ω 11 has been determined by analytical means and is presented in the Appendix. The resulting expansion given by (43) provides sufficient information to validate the numerical solutions obtained by numerical integration of equations (2) (21), which is the topic of the following section. 4. Numerical integration procedure. The numerical method implemented to solve equations (2) (21) is a spectral-finite difference scheme similar to that outlined in Staniforth [8]and will be briefly described below. We begin by discretizing the computational domain bounded by z z and θ 2π intoanetworkofn L grid points located at (48) (49) z i = ih, i =, 1,,N θ j = jk, j =, 1,,L with (5) (51) h = z N k = 2π L. Here, z refers to the outer boundary approximating infinity. By placing z well outside the growing boundary layer, this enables us to

12 226 S.J.D. D ALESSIO AND F.W. CHAPMAN enforce the far-field condition along the line z = z. We point out that the physical coordinate ξ = λz is a moving coordinate and hence the outer boundary ξ = λz is constantly being pushed further away from the cylinder surface at a rate which reflects the growth of the boundary layer. For this reason we are justified in saying that the vorticity, by the mechanism of convection, does not reach the outer boundary ξ. The stream function is expanded in a Fourier series given by (52) Ψ(z, θ, t) = 1 2 F (z, t)+ [F n (z, t)cosnθ + f n (z, t)sinnθ]. n=1 This transforms equation (2) into the following sets of equations for F n and f n,whichatagiventimet can be viewed as ordinary differential equations: (53) 2 f n z 2 n2 λ 2 f n = r n (z, t); n =1, 2, (54) 2 F n z 2 n2 λ 2 F n = s n (z, t); n =, 1,. The functions r n (z, t) ands n (z, t) are defined as (55) (56) r n (z, t) = 1 π s n (z, t) = 1 π 2π 2π M 2 ω sin nθ dθ; n =1, 2, M 2 ω cos nθ dθ; n =, 1,. Boundary conditions for F n and f n can be obtained by the no-slip condition which can be restated as (57) Ψ =, Ψ =onz =, z and the far-field conditions (14) which now become (58) λz Ψ e z 1 2 teξ sin(θ + α), λz Ψ e θ 1 2λ teξ cos(θ + α), ω as z.

13 INITIAL FLOW PAST A CYLINDER 227 Periodicity is automatically satisfied by the expansion. Hence, we find that (59) F (,t)=, F n (,t)=, f n (,t)= (6) F z =, F n z =, f n z = onz =forallt (61) e λz F, e λz F n 1 2λ teξ sin αδ 1,n, e λz f n 1 2λ teξ cos αδ 1,n as z and e λz F, F (62) z e λz n z 1 2 teξ sin αδ 1,n, e λz f n z 1 2 teξ cos αδ 1,n as z for n =1, 2,. The integral conditions can be formulated in terms of r n (z, t) ands n (z, t) as follows: (63) (64) (65) e nλz r n (z, t) dz = te ξ cos αδ 1,n e nλz s n (z, t) dz = te ξ sin αδ 1,n z s (z, t) dz =. Equations (53) (54) at a fixed value of t are of the form (66) h (z) β 2 h(z) =g(z) where β = nλ and the prime refers to differentiation with respect to z. These ordinary differential equations can be integrated using step-bystep formulae. The important point to note here is that the particular marching algorithm to be used is dependent on the parameter β. Dennis

14 228 S.J.D. D ALESSIO AND F.W. CHAPMAN and Chang [11]have found that most step-by-step procedures become increasingly unstable as β becomes large. Hence, two sets of step-bystep methods were utilized: one for β<.5and another one for β.5. The specific schemes used will not be presented; however, these details can be found in Staniforth [8]. The vorticity transport equation (21) is solved by finite differences. Previous studies for the case of a circular cylinder have also expanded the vorticity in a Fourier series; in that instance, the metric given by M 2 = e 2ξ is independent of θ. In our problem, the metric given by (9) depends on θ and consequently complicates matters greatly if expressed in a Fourier series. Despite this, Patel [12]for the case of a stationary inclined elliptic cylinder expanded the vorticity in a Fourier series and numerically solved the resulting partial differential equations for the Fourier coefficients. The scheme used to approximate equation (21) is very similar to the Crank-Nicolson implicit procedure. Equation (21) may be rewritten in the form (67) t ω t where (68) q(z, θ, t) = 1 2 ω 4M 2 z 2 +z 2 = q(z, θ, t) ω z +ω 2 + λ2 2 ω 4M 2 θ 2 + t ( Ψ ω M 2 θ z Ψ z ) ω. θ It is the finite-difference approximation to the time derivative that enables us to advance the solution step-by-step in time. Assuming the solution at time t is known, let us advance the solution to time t + t by integrating equation (67). Integration by parts yields (69) ωτ t+ t t t+ t t ωdτ = t+ t t qdτ where t is the time increment. If we approximate the integrals using the trapezoidal rule this brings us to the expression ( ) t (7) ω(z, θ, t+ t) =ω(z, θ, t)+ [q(z, θ, t+ t)+q(z, θ, t)]. 2t + t

15 INITIAL FLOW PAST A CYLINDER 229 Since q(z, θ, t + t) depends on ω(z, θ, t + t) and its derivatives, the scheme is implicit. Equation (7) is solved iteratively using the Gauss- Seidel procedure where all spatial derivatives appearing in (68) are approximated using central differences. The boundary conditions used in solving the vorticity transport equation include (71) ω(z, θ, t) =ω(z, θ +2π, t) and (72) ω(z,θ,t)=. The surface vorticity is determined by inverting (55) (56); this leads to the following expression (73) ω(,θ,t)= 1 { 1 M 2 2 s (,t)+ } [r n (,t)sinnθ+s n (,t)cosnθ]. n=1 To initiate the integration procedure, the solution at t =mustbe known. Setting t = and hence λ = in the series solution developed in the previous section yields the trivial solutions Ψ =, ω =. It may also be necessary to subject the surface vorticity to underrelaxation in order to obtain convergence. Convergence is reached when the difference between two successive iterates of the surface vorticity, ω (k+1) (,θ,t) ω (k) (,θ,t), falls below some specified tolerance ε. Lastly, we point out that the integrals appearing in (63) (64) were evaluated by Filon integration in order to guarantee consistent accuracy for all n. This technique bears a close resemblance to Simpson s rule with the exception that only the unknown numerically determined part of the integrand is approximated by a parabola over three successive grid points rather than the entire integrand. 5. Results and comparisons. The flow is characterized by the following dimensionless parameters: R, r and α. Numerical simulations were conducted for the cases R = 1 and R =1, with α =45 for an ellipse having r =.5.

16 23 S.J.D. D ALESSIO AND F.W. CHAPMAN To confirm numerical convergence, computations were carried out on two different grids: a fine grid having N L = , and an intermediate grid with N L = Results obtained from these computations showed that little was gained in going from the intermediate grid to the finer grid; thus, all results to be presented were generated using the intermediate grid. Other computational parameters used include an outer boundary of z =8,arelaxation parameter in the range.5 κ.75, 2 terms of the Fourier series, and a typical tolerance value of ε =1 6 was used. The number of terms retained in the series was suggested by the analytical solution, see the Appendix, where it was determined that 2 terms were sufficient in constructing the initial solution. This was also found to be adequate for all times considered in this study. Initial time steps of 1 4 were used for the first 1 advances. Then, the next 1 time steps were proceeded with t =1 3 and continued after with t =1 2. No stability difficulties were encountered with the choice of grids and parameters listed above. The numerical scheme was further tested by comparing the solutions with the analytical results previously derived and forms the main focus of this study. Of particular importance is the determination of the horizontal drag and vertical lift coefficients, C D and C L respectively, and their variation with time. The dimensionless drag and lift coefficients, obtained by integrating the pressure (P) and frictional stresses (F) on the surface, were computed using the formulae X P = 2sinhξ R X F = 2coshξ R Y P = 2coshξ R 2π ( ) ζ sin(θ) dθ ξ 2π 2π 2π ζ sin(θ) dθ ( ) ζ cos(θ) dθ ξ Y F = 2sinhξ R ζ cos(θ) dθ. Defining X = X P + X F and Y = Y P + Y F we then arrive at (74) C D = X cos(α) Y sin(α) (75) C L = Y cos(α)+x sin(α).

17 INITIAL FLOW PAST A CYLINDER 231 FIGURE 1a. Comparison of the time variation of the drag coefficient for the case R = 1, r =.5 andα =45. Using the approximation given by (43) for the vorticity, this enables us to determine an expression for the drag and lift coefficients which will be valid for small times following the start of the motion. The surface vorticity and its normal derivative needed to compute the drag FIGURE 1b. Comparison of the time variation of the lift coefficient for the case R = 1, r =.5 andα =45.

18 232 S.J.D. D ALESSIO AND F.W. CHAPMAN and lift coefficients are given approximately by (76) [ ζ = ζ(,θ,t) teξ 4sin(θ + α) sinh(2ξ ] )sin(θ + α) M πλ 4M 3 + (a n sin(nθ)+b n cos(nθ))] n=1 (77) ( ) [ ζ Reξ sin(θ +α)+ λ ξ 2 π n=1 ] (a n sin(nθ)+b n cos(nθ)) where a n and b n are defined by (A6) in the Appendix. The special case of a circular cylinder can be obtained by letting ξ and α =. The surface vorticity given by (76) then becomes ζ 8t πλ sin θ, which is in full agreement with the first term determined by Collins and Dennis [1]. The early time expressions for the drag and lift then become (78) C D = d + d 1 t + d2 t C L = l + l 1 t + l2 t where the constants d and l, corresponding to the drag and lift at t = respectively, have been determined to be d = π(sin 2 α + e ξ sinh ξ ) and l = π sin α cos α and depend only on the inclination and aspect ratio. The remaining constants have been numerically determined. For R = 1 these values are d 1 = , d 2 =.62774, l 1 =.32412, l 2 = while for R =1, these constants assume the values of d 1 = , d 2 =.6277, l 1 =.95631, l 2 = Contrasted in Figures 1a,b are the computed drag and lift coefficients against the formulae given by (78) for R =1,. The plots reveal that the agreement is quite good until a time t =.5; afterwards the deviation increases as expected. Similar agreement is observed

19 INITIAL FLOW PAST A CYLINDER 233 FIGURE 2a). Comparison of the time variation of the drag coefficient for the case R = 1, r =.5 andα =45. in Figures 2a,b, which compare C D and C L for the case R = 1. Also contrasted are the surface vorticity distributions which are shown in Figures 3a e corresponding to the times t =.1,.2,.3,.4,.5, respectively. As noted in the plots, the surface vorticity has been scaled by the factor R. These plots illustrate how the agreement between the numerical solution and equation (76) is excellent for small times and worsens as time progresses. Further, the importance of the second term in the series (43), given by ω 11, is shown to contribute only a small FIGURE 2b). Comparison of the time variation of the lift coefficient for the case R = 1, r =.5 andα =45.

20 234 S.J.D. D ALESSIO AND F.W. CHAPMAN correction to the first term, ω 1. This is confirmed in Figures 4a and b which show the surface vorticity distribution using the first term and both terms together. We add that although the correction is small, it acts so as to improve the overall agreement with the numerically obtained distribution. Revealed in these plots is the rapid variation in vorticity near the tips of the ellipse. Since the drag and lift coefficients are related to integrals of the surface vorticity and its normal derivative, the agreement between the numerical and approximate solutions takes place over a larger time interval. It is well established that for large Reynolds numbers the pressure contribution to the drag and lift dominates over the frictional component. An important measure of the pressure force is obtained from the dimensionless (viscous) pressure coefficient, defined as (79) P (ξ =,θ,t)=p (,θ,t) P (,,t)= 2 θ ( ) ζ d θ. R ξ Using our analytical expression given by (77), a formula for P valid for small times can be derived. The resulting formula takes the form (8) P (,θ,t) e ξ [cos(θ + α) cos α] eξ λ π n=1 [ an n (1 cos(nθ)) + b ] n n sin(nθ). Further, a corresponding expression can be determined for the inviscid case from potential flow theory and is outlined in the Appendix. This yields (81) P (,θ,t)=e ξ [cos(θ + α) cos α] t 2 e 2ξ θ sin( θ + α) M 2 ( θ) [ cos( θ + α) sin(2 θ)sin( θ ] + α) 2M 2( θ) d θ. Since (8) is expected to be in close agreement with the corresponding numerical values for small times we contrast the viscous (8) and inviscid (81) pressure coefficients for t =.1, t =.2 in Figures 5a and b, respectively, for R = 1,. The agreement illustrated verifies how the inviscid theory is successful in predicting the main features for this

21 INITIAL FLOW PAST A CYLINDER 235 FIGURE 3a. Comparison of the surface vorticity distribution for the case R = 1, r =.5 andα =45 at time t =.1. case. In Figure 6 a similar comparison is drawn for t = 1; here, however, the viscous case is obtained from the numerical solution by using (79) since (8) is no longer expected to hold. We emphasize though that as t increases the departure between the viscous and inviscid solutions will become significant due to boundary-layer separation. FIGURE 3b. Comparison of the surface vorticity distribution for the case R = 1, r =.5 andα =45 at time t =.2.

22 236 S.J.D. D ALESSIO AND F.W. CHAPMAN FIGURE 3c. Comparison of the surface vorticity distribution for the case R = 1, r =.5 andα =45 at time t =.3. Lastly, we present streamline plots of the flow at times t =1andt =2 in Figures 7a and b, respectively, for the case R = 1, α =45 and r =.5. Figures 8a and b correspond to similar flow patterns for the case R =1,. The plots indicate that for R =1,, a vortex has been shed from the trailing tip at some time between t =1andt =2. This is evidenced by the closed streamline shown near the trailing tip of the cylinder. For R = 1, vortex shedding does not appear to occur in the time interval t<2. As a final note we wish to point out that FIGURE 3d. Comparison of the surface vorticity distribution for the case R = 1, r =.5 andα =45 at time t =.4.

23 INITIAL FLOW PAST A CYLINDER 237 FIGURE 3e. Comparison of the surface vorticity distribution for the case R = 1, r =.5 andα =45 at time t =.5. the spacing between consecutive streamlines decreases as t increases because the far-field velocity is steadily increasing with time. 6. Conclusions. Considered in this paper was the unsteady twodimensional flow of a viscous incompressible fluid past a uniformly accelerating elliptic cylinder starting from rest. To adequately model the evolving early flow development and to sufficiently resolve the thin FIGURE 4a. Comparison of the surface vorticity distribution for the case R = 1, r =.5 andα =45 at t =.1 using one term of the series.

24 238 S.J.D. D ALESSIO AND F.W. CHAPMAN FIGURE 4b. Comparison of the surface vorticity distribution for the case R = 1, r =.5 andα =45 at time t =.1 using two terms of the series. boundary layer, a transformation was introduced. This boundary-layer type transformation incorporated the known early structure of the flow into the equations of motion. A numerical technique involving both finite difference and spectral methods was described and was successful in computing the early stages of the flow following the start of the motion for moderate to large Reynolds numbers. FIGURE 5a. Comparison of the inviscid and viscous pressure coefficients at time t =.1 for the case R = 1, r =.5 andα =45.

25 INITIAL FLOW PAST A CYLINDER 239 FIGURE 5b. Comparison of the inviscid and viscous pressure coefficients at time t =.2 for the case R = 1, r =.5 andα =45. Also presented was an approximate series solution expressed in powers of the time t and the parameter λ = 8t/R. It was shown by comparing the approximate solution against numerical solutions that the series was valid for small times and for moderately large Reynolds numbers. For early times the two forms of solutions were found to be in excellent agreement and as time progressed the agreement worsened. In FIGURE 6. Comparison of the inviscid and viscous pressure coefficients at time t =1forthecaseR = 1, r =.5 andα =45.

26 24 S.J.D. D ALESSIO AND F.W. CHAPMAN FIGURE 7a. time t =1. Streamline plots for the case R = 1, r =.5 andα =45 at FIGURE 7b. time t =2. Streamline plots for the case R = 1, r =.5 andα =45 at FIGURE 8a. t =1. Streamline plots for R = 1, r =.5 andα =45 at time

27 INITIAL FLOW PAST A CYLINDER 241 FIGURE 8b. t =2. Streamline plots for R = 1, r =.5 andα =45 at time particular, two cases were considered for an ellipse inclined at 45 with the horizontal and having r =.5: R = 1 and R =1,. Lastly, instantaneous streamline plots were presented and some comparisons were made against the potential flow solution. Future work will involve the problem whereby the cylinder is undergoing both uniform acceleration as well as harmonic oscillations in its inclination. This can be used as a model to simulate an accelerating airfoil passing through turbulence. Appendix Determination of ω 11. The analytical determination of the term ω 11 is outlined below. We begin by defining (A1) x = M z, Ω 11 = M ω 11, Ω 1 = M ω 1. Using the known solution for ω 1 equation (45) becomes 2 Ω 11 (A2) x 2 +2x Ω 11 x 4Ω sinh(2ξ 11 = 8eξ ) πm 3 xe x2 sin(θ + α). The homogeneous solution Ω 11,h of (A2) can be easily verified to be x (A3) Ω 11,h (x, θ) =A(θ)(2x 2 +1)+B(θ)(2x 2 e t2 dt +1) (2t 2 +1) 2 where A and B are arbitrary functions of θ which must be determined. From the two linearly independent solutions found in (A3), the particular solution Ω 11,p can be constructed by the method of variation of

28 242 S.J.D. D ALESSIO AND F.W. CHAPMAN parameters and expressed in a simplified convenient form given by (A4) Ω 11,p (x, θ) = eξ sinh(2ξ ) 2M 3 x (2x 2 +1)erfc(x)sin(θ + α). Imposing the condition that Ω 11 as x yields A(θ) = πb(θ)/4 since e t2 dt π (2t 2 +1) 2 = 4. With this, the complete solution for Ω 11 (x, θ) can be written compactly as (A5) ( Ω 11 (x, θ) = (2x 2 e t2 dt +1) B(θ) (2t 2 +1) 2 + eξ sinh(2ξ ) 2M 3 ) erfc (x)sin(θ + α). The unknown function B(θ) is lastly determined by satisfying the integral conditions (47) and making use of the results (2x 2 +1) x e t2 dt (2t 2 +1) 2 dx = 1 6 and (2x 2 +1)erfc(x) dx = 5 3 π. The resulting expression for B(θ) is given by the following Fourier series (A6) B(θ) = eξ sinh(2ξ )sin(θ + α) πm 3 4eξ [a n sin(nθ)+b n cos(nθ)] π n=1 where and a n = n π b n = n π 2π 2π sin(φ + α)sin(nφ) M (φ) sin(φ + α)cos(nφ) M (φ) dφ dφ.

29 INITIAL FLOW PAST A CYLINDER 243 The Fourier coefficients a n and b n were found to diminish rapidly with n. Only 2 terms are needed in the Fourier series to yield six decimal place accuracy. This concludes the solution procedure for determining ω 11 (x, θ) = Ω 11 (x, θ)/m (θ). The corresponding stream function Ψ 11 can, in principle, be found by solving (44) subject to (46) since both ω 1 and ω 11 are now known. Because we are only interested in comparing surface vorticity distributions and drag and lift coefficients, the solution of Ψ 11 was not pursued. As a final note, some of the intermediate steps in the analytical solution were performed with assistance of the Maple computer algebra system. Potential flow solution. The dimensionless inviscid equations in vector form are V (A7) ( t = P + 1 ) V 2 V where V =(v ξ,v θ ). The θ-component of the above equation becomes (A8) v θ t = 1 M ( P + 1 ) θ 2 [v2 ξ + vθ] 2. On the cylinder surface the impermeability condition gives v ξ =,and (A8) simplifies to ( ) P (A9) = 1 [ ] ( ) vθ θ 2 θ (v2 θ) M. t The velocity component v θ is determined from the stream function through equation (1). From potential flow theory, ζ =, and hence the stream function satisfies (A1) 2 ψ ξ ψ θ 2 =. The solution to (A1) subject to ψ =, ψ θ = onξ =

30 244 S.J.D. D ALESSIO AND F.W. CHAPMAN and is ψ t 2 eξ +ξ sin(θ + α) as ξ (A11) ψ(ξ,θ,t) =te ξ sinh ξ sin(θ + α). Thus, we find that (A12) v θ (,θ,t)= teξ M sin(θ + α), which is in full agreement with equation (39), as expected. Substituting (A12) into (A9) and integrating we then arrive at the final result for the inviscid pressure coefficient: (A13) P (,θ,t)=p (,θ,t) P (,,t) = e ξ [cos(θ + α) cos α] t 2 e 2ξ θ sin( θ + α) M 2 ( θ) [ cos( θ + α) sin(2 θ)sin( θ + α) 2M 2 ( θ) ] d θ. Acknowledgment. Financial support for this research was provided by the Natural Sciences and Engineering Research Council of Canada. REFERENCES 1. W.M. Collins and S.C.R. Dennis, Symmetrical flow past a uniformly accelerated circular cylinder, J.FluidMech.65 (1974), H.M. Badr, S.C.R. Dennis and S. Kocabiyik, Symmetrical flow past an accelerated circular cylinder, J. Fluid Mech. 38 (1996), H.M. Badr and S.C.R. Dennis, Time-dependent viscous flow past an impulsively started rotating and translating circular cylinder, J. Fluid Mech. 158 (1985), S.C.R. Dennis and L. Quartapelle, Some uses of Green s theorem in solving the Navier-Stokes equations, Internat. J. Numer. Meth. Fluids 9 (1989), H. Blasius, Grenzschichten in Flüssigkeiten mit Kleiner Reibung, ZAMP56 (198), (English Translation NACA TM 1256.) 6. W.M. Collins and S.C.R. Dennis, The initial flow past an impulsively started circular cylinder, Quart. J. Mech. Appl. Math. 26 (1973),

31 INITIAL FLOW PAST A CYLINDER , Flow past an impulsively started circular cylinder, J. Fluid Mech. 6 (1973), A.N. Staniforth,, Ph.D. Thesis, University of Western Ontario, London, Canada, S.C.R. Dennis and A.N. Staniforth, A numerical method for calculating the initial flow past a cylinder in a viscous fluid, in Proc. 2nd Internat. Conf. Num. Meth. Fluid Dyn. (M. Holt, ed.), Lecture Notes in Physics, vol. 8, Springer-Verlag, Berlin, 1971, pp S.J.D. D Alessio, S.C.R. Dennis and P. Nguyen, Unsteady viscous flow past an impulsively started oscillating and translating elliptic cylinder, J.Engrg.Math. 35 (1999), S.C.R. Dennis and Gau-Zu Chang, Numerical integration of the Navier- Stokes equations in two dimensions, Mathematics Research Center, University of Wisconsin, Technical Summary Report No. 859, 1969, 89 pp. 12. V.A. Patel, Flow around the impulsively started elliptic cylinder at various angles of attack, Comp. Fluids 9 (1981), Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 address: sdalessio@math.uwaterloo.ca Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 address: fwchapman@daisy.uwaterloo.ca