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The stability of a transonic boundary layer on an elastic surface
Journal of Applied Mathematics and Mechanics 75 (2011) 357–362
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The stability of a transonic boundary layer on an elastic surface夽 A.N. Bogdanov, V.N. Diyesperov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 24 May 2010
a b s t r a c t The perturbations in the boundary layer over an elastic surface when there is non-stationary free viscousinviscid interaction at transonic velocities are investigated using a modified three-deck model. The modification consists of retaining the term with the second derivative with respect to time (the singular term of the transonic expansion), which occurs in the model of the Lin–Reissner–Tsien equation when it is derived from the complete equations for the velocity potential. This enables the equations of the model to be improved so that they more accurately describe non-stationary and non-linear phenomena. It is shown that the modified model enables perturbations, ignored when using the classical three-deck model, to be taken into account. The compliance on the surface may lead to a reduction in the perturbation growth rate. © 2011 Elsevier Ltd. All rights reserved.
An investigation of the development of perturbations of the boundary layer in transonic flow has shown1 that the elasticity of the surface over which the flow occurs has a considerable influence on the growth and damping of the perturbations. There are experimental results, which show a reduction in the drag of bodies, whose surface contains elastic components.2,3 One of the possible explanations for this effect is the reduction in the perturbations and the stabilization of the boundary layer on the surface of such a body.4,5 The research described in Ref. 1 was carried out using a three-deck model of non-stationary transonic free viscous-inviscid interaction.6 A drawback of that investigation is that it is unable to give a complete picture of the distribution of the non-stationary perturbations.7 In this paper we therefore use a modification of the three-deck model.7 The modification consists of retaining the term with the second derivative with respect to time, which occurs in the model of the Lin–Reissner–Tsien (LRT) equation, when it is derived from the complete equations for the velocity potential. The equation obtained in this way can be usefully called the modified LRT equation. Unlike the usual LRT equation8 it describes the propagation of non-stationary perturbations in the flow field in any direction. The use of this model when investigating a number of problems of non-stationary free viscous-inviscid interaction at transonic velocities enables the features of the development of non-stationary perturbations to be described more exactly. It was shown in Refs 7 and 9 that there are perturbations in the boundary layer that are not described by the usual three-deck model. 1. Derivation of the dispersion relation We will use the modified three-deck model.7 We will choose as the main flow a one-dimensional steady flow along the x axis, directed along the surface of a plane plate. In the region of the surface over which the flow occurs in the three-deck model, the equations have the usual form of the equations of the non-stationary boundary layer for an incompressible fluid (1.1) where u and v are the velocity components along Cartesian coordinate axes x and y, and p is the pressure. Inviscid transonic flow can be approximately assumed to be vortex-free,8 far from the surface over which the flow occurs, and to describe it we will use the modified linear Lin–Reissner–Tsien equation7 ((t, x, y) is the velocity potential and M∞ is the Mach number of the unperturbed flow) (1.2)
夽 Prikl. Mat. Mekh. Vol. 75, No. 3, pp. 505–512, 2011. E-mail address: [email protected] (A.N. Bogdanov). 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.07.014
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The term with the second derivative with respect to time in this equation is singular. This term is omitted in the usual Lin–Reissner–Tsien equation, but in this case the model describes non-stationary perturbations, which only propagate upstream.8 The derivation of the equations for the middle deck and their integration with the corresponding boundary conditions were given in Refs 1, 6. Here we will only give the corresponding equations (matching conditions), which follow from the general analysis, in the case of flow over an elastic surface1 (1.3) (1.4) where A(t, x) is a function of the instantaneous displacement of the transition-flow streamlines and y = F(t, x) is the law of deformation of the elastic surface. Note that, in the three-deck model, a transformation of the spatial coordinates and time is used, such that the x coordinate and the time t are the same for all the decks, and the y1 and y coordinates have different scales (since the thickness of the boundary layer and of the transition flow are different). We will use the condition that the solution is bounded at the exit from the region being investigated
For the components of the flow velocity on the plate surface we have, by the no-slip and impermeability conditions, (1.5) We will choose the relation between the deformation of the elastic surface and the pressure in the form (g is the coefficient of elasticity)1 (1.6) We will investigate the behaviour of small perturbations of the amplitude ␦˜ 1. To be specific, we will choose the linear profile of the unperturbed flow u = y for constant pressure. Suppose the initial conditions have the form (1.7) and there are no incoming perturbations (1.8) Representing the parameters of the perturbed flow in the form of series in powers of ␦˜
and substituting them into system (1.6), we have (1.9) From the initial and boundary conditions (1.4), (1.5), (1.7) and (1.8) we obtain
(1.10) ¯ The stability of the freely interacting boundary layers have been traditionally investigated for perturbations of the form u(y) exp(t + ikx), where the detection of cases where Re > 0 indicates flow instability. This problem is similar to finding solutions of the linearized system of basic equations using a Fourier transformation with respect to x and a Laplace transformation with respect to t. Carrying out these transformations, we can reduce system (1.9) to a single equation for the transform u1 , by differentiating the last equation of system (1.9) with respect to y and eliminating v¯ 1 , using the first equation of this system. We obtain
(1.11) Three boundary conditions are necessary to solve Eq. (1.11). The first of these can be obtained as a result of Fourier and Laplace transformations of the corresponding condition (1.10): (1.12) The second is derived as a result of Fourier and Laplace transformations of the last equation of (1.9). Taking the equalities u¯ 1 = v¯ 1 = 0 into account we have (1.13) The third boundary condition represents the transformation of the last condition of (1.10): (1.14)
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Introducing the new independent variable
we reduce Eq. (1.11) to an Airy equation10 in du¯ 1 /dz. The solution of this equation can be expressed in terms of the one of the fundamental solutions – the Airy function Ai(z) – as follows (1.15) The second is a linearly independent solution – a function that is unbounded as z → ∞ and must be dropped. In turn, the solution of Eq. (1.15) has the form
(1.16) Boundary conditions (1.12) and (1.14), on changing to the new variable z, remain unchanged and are now satisfied when z = and z → ∞, respectively. Boundary condition (1.13) becomes the condition (1.17) Corresponding to condition (1.12), taking into account the remarks made on changing to the variable z, we have C1 = 0 for solution (1.16). Substituting solution (1.16) into condition (1.17), we obtain (1.18) Similarly, we have from condition (1.14)
(1.19) It follows from Eqs (1.18) and (1.19) that
(1.20) ¯ we obtain Carrying out Fourier and Laplace transformations of the matching conditions (1.3) and (1.4) for p¯ 1 and A, (1.21) The transform ¯ is found from the transformed equation (1.2) (1.22) The solution of Eq. (1.22) has the form (1.23) ¯ is bounded as y1 = ∞. In the solution obtained we have chosen the minus sign, which ensures that Substituting solution (1.23) into relations (1.21) and eliminating , we obtain (1.24) Eliminating the quantities A¯ and P¯ 1 from Eqs (1.20) and (1.24) and using the Fourier and Laplace transformed expression (1.6), we obtain the dispersion relation
(1.25) Taking the limit in the case of a rigid wall (g → ∞) in dispersion relation (1.25) we get the dispersion relation obtained previously7
when investigating the stability of the boundary layer in the case of transonic flow over a rigid plate. When ␦ = 0, dispersion relation (1.25) gives the dispersion relation obtained previously in Ref. 11 when investigating transonic flow over an elastic surface in the classical three-deck model. The previous analysis of the dispersion relation containing the combination I(), showed (see, for example, Ref. 11), that dispersion relations of this kind have an infinite number of solutions. It can be seen that the inclusion of the elasticity of the surface over which the flow occurs does not increase the number of perturbation waves in the boundary layer compared with the case of flow over a rigid wall (formally due to the fact that the effect of elasticity is described by terms of the dispersion relation (1.25) which depend on g of lower order of smallness in , that already exist in them). The form of dispersion relation (1.25) enables us to express qualitative considerations regarding the effect of the elasticity of the surface over which the flow occurs, based on the fact that a formal consideration of the elasticity leads to the occurrence, in the left-hand side
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of this dispersion relation, of a term with real and imaginary parts, which depend only on k. Earlier investigations1,6 showed a nonmonotonic behaviour of both and as k increases. Consequently, there are ranges of wave numbers k, for which there will be a value of , corresponding to the chosen value of the wave number k, that is less than when flow takes place over a rigid plate, and this denotes a lower rate of change of the perturbation with time (a smaller increase and a slower damping). Analytically, the roots of dispersion relation (1.25) are easier to find in the plane. Since
then, if we assume k to be a positive √ real number, when Re ˝ > lie below the straight line Re = 3Im.
√
3Im ˝ the roots increase with time Re > 0; consequently, unstable roots
2. The asymptotic of the dispersion relation as k → 0 The search for the relation = (k) can be organized as follows (arising from previous investigations of the stability of interacting boundary layers* :1,6,11 we choose as initial points values of for k = 0. For a zero right-hand side of dispersion relation (1.25) the zeros of its left side occur at the zeros of the derivative of the Airy function, since the integral of the Airy function in the limits from to ∞ takes finite values for any values of . In turn, the derivative of the Airy function only has zeros on the negative part of the real axis.10 There is a denumerable number of such points, and they can be determined with high accuracy by the following formula (m is a natural number)
The asymptotic of the left-hand side of dispersion relation (1.25) can be obtained, for large negative values of , using the asymptotic of the derivative of this function10 for large negative values of the argument. Dropping the first term of this asymptotic, we obtain11
which enables us11 to construct the asymptotic of the roots of dispersion relation (1.25) as k → 0
When k → 0 the zeros of the left-hand side of dispersion relation (1.25) are determined not only by the zeros of the derivative of the Airy function, since there is already one zero: 0 = 0 (it also occurs when ␦ = 0 (Ref. 1). This root is not identical zero: it can be seen from dispersion relation (1.25) that it has no solutions 0 ≡ 0, since for any k it obviously does not satisfy the equality
For the calculated derivative ∂/∂k, for all cases, apart from K∞ = 0, by virtue of dispersion relation (2.7) when k → 0, = 0, we have
Hence, from the start of the change in k from zero, the root 0 departs from the origin of coordinates. A similar result is also obtained in the case when ␦ = 0. However, the asymptotic of the root 0 when 0 → 0 in the cases when ␦ = 0 and ␦ = / 0 is different. For small values of the argument the functions Ai = Ai() we have10
Confining ourselves, in the expansion of the function I(), to the term independent of , we obtain from dispersion relation (1.25) an equation which has two roots. Returning to the variable , we write these roots in the form
(2.1) The first of the roots (2.1) in the principal terms in k is identical with that obtained earlier in Ref. 1. It is stable, since
which, in the case considered, gives a negative value of Re. The second root has a singular pure imaginary term and a sign of the regular part that is opposite to that of the first root; obviously, in the case considered this root increases. When ␦ = 0 the second root drops out and is therefore not described by the classical theory.
∗
Terentev YeD. Non-stationary boundary-layer problems with free interaction. Doctorate Dissertation. Computational Centre, Russian Academy of Sciences, 1986.
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Hence, for small k and → 0, the elasticity of the surface over which the flow occurs leads to damping (growth) of the perturbation, corresponding to the first (second) root, whereas for flow over a solid surface, neutral perturbations correspond to both roots (in this approximation). As k → ∞, the integral of the Airy function mainly defines the asymptotic of the dispersion relation (unlike the asymptotic when k → 0). That is, for dispersion relation (1.25) as k → ∞ the dispersion curves, emerging from the zeros of the arbitrary Airy function, pass through points defined by the zeros of the equation
Their asymptotic representation for large m is as follows:12
We can say once again, that the root of dispersion relation (1.25), which departs to infinity, is determined by the asymptotic ratio of the derivative of the Airy function and the integral of this function, namely (2.2) The asymptotic behaviour of dispersion relation (1.25) as k → ∞ also depends on the asymptotic behaviour of the factor with the square root; it defines the asymptotic of the additional root * as k → ∞. In order to obtain correspondence with the results obtained earlier7 when g → ∞, we also retain in the expansion of dispersion relation (1.25) terms independent of g:
(2.3) When solving relation (2.3), it is much more convenient to convert it into a fourth-degree algebraic equation in : (2.4) It can be seen that when ␦ = 0, the leading term of Eq. (2.4) falls out and the order of the equation is reduced by one. Consequently, when ␦ = / 0 there is an additional root. This root probably corresponds to the wave that arises when there is interaction between the perturbation of the external inviscid flow, propagating downstream with the boundary layer and is special in the sense that it is not described by the classical three-deck model. As can be seen from dispersion relation (1.25), this special root exists independently of the number of terms remaining in expansion (2.2) The case ␦ = 0 was investigated previously and the following high-frequency asymptotic was obtained1 (2.5) Taking the limit of a rigid wall g → ∞ in asymptotic (2.5) is not feasible, since when g → ∞ we have → ∞ and the statement that the roots of the dispersion relation in the case considered and in the case of a rigid surface correspond to one another is impossible. Nevertheless, correspondence should occur since dispersion relation (1.25) when g → ∞ becomes the dispersion relation for the case of flow over a rigid wall.7 Since it is necessary to indicate this correspondence of the roots, below when g → ∞, to fix our ideas, we will assume that g 1 and k/g → 1. Assuming k is a real number, it is more convenient to convert Eq. (2.4) into an equation with real coefficients. When ␦ = 0 it reduces to a third-order equation (2.6) which has three roots; in the principal terms in k when k → ∞ they can be written in the form (2.7) When g → ∞ the asymptotic 1 = k5/3 + 5kKˆ ∞ + . . . , was obtained in Ref. 6, which is almost identical with the first of the roots of (2.7). On the other hand, it can be seen that the term k5/3 in asymptotic (2.5) is not present. Dispersion relation (2.6) has a singular root (2.8) ∗ K∞
−k2 g 2
2 − ␦ k2
− 2␦k2 g −1
if K∞ = ≡ ∗ , for which this root This root, in the case of flow over a rigid wall (g → ∞), was obtained in Ref. 7 (it differs only in the value of K∞ exists). The root obtained corresponds to the solution of a form of travelling wave and exists, as can be seen, for negative values of the ∗ (in the subsonic region of transonic flow). It can be seen that the elasticity of the wall leads to an increase in the transonic parameter K∞ ∗ (residual negative). When the coefficient of elasticity g decreases the parameter K ∗ increases in modulus modulus of the parameter K∞ ∞ (transition from the transonic range of velocities to the subsonic region).
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We can refine the asymptotic of the singular root (2.8):
∗ . which is a generalization to the case K∞ = / K∞ When ␦ = / 0 dispersion relation (2.4) can be solved by Ferrari’s method. Expanding in series in ␦ 1, in the principal terms in k as k → ∞ we have, in the notation of (2.6),
(2.9) Hence, all the roots are pure imaginary. When ␦ = 0 the last root of (2.9) drops out, and in this sense it is singular, like the case of nonstationary transonic aerodynamics, when one root drops out if the modelling of the propagation of weak non-stationary perturbations is carried out using the Lin–Reissner–Tsien equation.12 The remaining roots are regular in ␦. When g → ∞, the roots (2.9) acquire the form corresponding to the case of a rigid surface over which the flow occurs. The effect of the elasticity of the surface reduces to an increase by (k/g)2 of the value of the transonic parameter, for which development of the perturbations in the flow occurs: instead of flow with a subsonic or sonic velocity, weakly supersonic flow occurs (the reduction in g leads to the transition from transonic to supersonic velocity range). References 1. Savenkov IV. The effect of the elasticity of the surface over which flow occurs on the stability of the boundary layer at transonic velocities of the external flow. Zh Vychisl Mat Mat Fiz 2001;41(1):135–40. 2. Kramer MO. Boundary-layer stabilization by distributed damping. J Aeronaut Sci 1957;24(6):459–60. 3. Kramer MO. Boundary-layer stabilization by distributed damping. J Amer Soc Naval Engrs 1960;72:25–33. 4. Carpenter PW, Garrad AD. The hydrodynamic stability of flow over Kramer-type complaint surfaces Pt1. Tollmien–Schlichiting instabilities. J Fluid Mech 1985;155:465–510. 5. Carpenter PW, Garrad AD. The hydrodynamic stability of flow over Kramer-type complaint surfaces. Pt2. Flow-induced surface instabilies. J Fluid Mech 1986;170:199–232. 6. Ryzhov OS, Savenkov IV. The stability of the boundary layer at transonic velocities of the external flow. Zh Prikl Mekh Tekh Fiz 1990;2:65–71. 7. Bogdanov AN, Diyesperov VN. Modeling of non-stationary transonic flow and the stability of the transonic boundary layer. Prikl Mat Mekh 2005;69(3):394–403. 8. Cole,J.D., Cook, L.P. Transonic aerodynamics. Amsterdam: etc.: North Holland, 1986. 9. Bogdanov AN, Diyesperov VN. Tolmien–Schlichting waves in a transonic boundary layer Excitation from the external flow and from the surface. Prikl Mat Mekh 2007;71(2):289–300. 10. Abramowitz M, Stegun IA, editors. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables.. Washington: National Bureau of Standards; 1964. 11. Zhuk VI. Tolmien–Schlichting Waves and Solitons. Moscow: Nauka; 2001. 12. Bogdanov AN. Higher approximations of the transonic expansion in problems of unsteady transonic flows. Prikl Mat Mekh 1997;61(5):798–811.
Translated by R.C.G.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The stability of a transonic boundary layer on an elastic surface夽 A.N. Bogdanov, V.N. Diyesperov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 24 May 2010
a b s t r a c t The perturbations in the boundary layer over an elastic surface when there is non-stationary free viscousinviscid interaction at transonic velocities are investigated using a modified three-deck model. The modification consists of retaining the term with the second derivative with respect to time (the singular term of the transonic expansion), which occurs in the model of the Lin–Reissner–Tsien equation when it is derived from the complete equations for the velocity potential. This enables the equations of the model to be improved so that they more accurately describe non-stationary and non-linear phenomena. It is shown that the modified model enables perturbations, ignored when using the classical three-deck model, to be taken into account. The compliance on the surface may lead to a reduction in the perturbation growth rate. © 2011 Elsevier Ltd. All rights reserved.
An investigation of the development of perturbations of the boundary layer in transonic flow has shown1 that the elasticity of the surface over which the flow occurs has a considerable influence on the growth and damping of the perturbations. There are experimental results, which show a reduction in the drag of bodies, whose surface contains elastic components.2,3 One of the possible explanations for this effect is the reduction in the perturbations and the stabilization of the boundary layer on the surface of such a body.4,5 The research described in Ref. 1 was carried out using a three-deck model of non-stationary transonic free viscous-inviscid interaction.6 A drawback of that investigation is that it is unable to give a complete picture of the distribution of the non-stationary perturbations.7 In this paper we therefore use a modification of the three-deck model.7 The modification consists of retaining the term with the second derivative with respect to time, which occurs in the model of the Lin–Reissner–Tsien (LRT) equation, when it is derived from the complete equations for the velocity potential. The equation obtained in this way can be usefully called the modified LRT equation. Unlike the usual LRT equation8 it describes the propagation of non-stationary perturbations in the flow field in any direction. The use of this model when investigating a number of problems of non-stationary free viscous-inviscid interaction at transonic velocities enables the features of the development of non-stationary perturbations to be described more exactly. It was shown in Refs 7 and 9 that there are perturbations in the boundary layer that are not described by the usual three-deck model. 1. Derivation of the dispersion relation We will use the modified three-deck model.7 We will choose as the main flow a one-dimensional steady flow along the x axis, directed along the surface of a plane plate. In the region of the surface over which the flow occurs in the three-deck model, the equations have the usual form of the equations of the non-stationary boundary layer for an incompressible fluid (1.1) where u and v are the velocity components along Cartesian coordinate axes x and y, and p is the pressure. Inviscid transonic flow can be approximately assumed to be vortex-free,8 far from the surface over which the flow occurs, and to describe it we will use the modified linear Lin–Reissner–Tsien equation7 ((t, x, y) is the velocity potential and M∞ is the Mach number of the unperturbed flow) (1.2)
夽 Prikl. Mat. Mekh. Vol. 75, No. 3, pp. 505–512, 2011. E-mail address: [email protected] (A.N. Bogdanov). 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.07.014
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The term with the second derivative with respect to time in this equation is singular. This term is omitted in the usual Lin–Reissner–Tsien equation, but in this case the model describes non-stationary perturbations, which only propagate upstream.8 The derivation of the equations for the middle deck and their integration with the corresponding boundary conditions were given in Refs 1, 6. Here we will only give the corresponding equations (matching conditions), which follow from the general analysis, in the case of flow over an elastic surface1 (1.3) (1.4) where A(t, x) is a function of the instantaneous displacement of the transition-flow streamlines and y = F(t, x) is the law of deformation of the elastic surface. Note that, in the three-deck model, a transformation of the spatial coordinates and time is used, such that the x coordinate and the time t are the same for all the decks, and the y1 and y coordinates have different scales (since the thickness of the boundary layer and of the transition flow are different). We will use the condition that the solution is bounded at the exit from the region being investigated
For the components of the flow velocity on the plate surface we have, by the no-slip and impermeability conditions, (1.5) We will choose the relation between the deformation of the elastic surface and the pressure in the form (g is the coefficient of elasticity)1 (1.6) We will investigate the behaviour of small perturbations of the amplitude ␦˜ 1. To be specific, we will choose the linear profile of the unperturbed flow u = y for constant pressure. Suppose the initial conditions have the form (1.7) and there are no incoming perturbations (1.8) Representing the parameters of the perturbed flow in the form of series in powers of ␦˜
and substituting them into system (1.6), we have (1.9) From the initial and boundary conditions (1.4), (1.5), (1.7) and (1.8) we obtain
(1.10) ¯ The stability of the freely interacting boundary layers have been traditionally investigated for perturbations of the form u(y) exp(t + ikx), where the detection of cases where Re > 0 indicates flow instability. This problem is similar to finding solutions of the linearized system of basic equations using a Fourier transformation with respect to x and a Laplace transformation with respect to t. Carrying out these transformations, we can reduce system (1.9) to a single equation for the transform u1 , by differentiating the last equation of system (1.9) with respect to y and eliminating v¯ 1 , using the first equation of this system. We obtain
(1.11) Three boundary conditions are necessary to solve Eq. (1.11). The first of these can be obtained as a result of Fourier and Laplace transformations of the corresponding condition (1.10): (1.12) The second is derived as a result of Fourier and Laplace transformations of the last equation of (1.9). Taking the equalities u¯ 1 = v¯ 1 = 0 into account we have (1.13) The third boundary condition represents the transformation of the last condition of (1.10): (1.14)
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359
Introducing the new independent variable
we reduce Eq. (1.11) to an Airy equation10 in du¯ 1 /dz. The solution of this equation can be expressed in terms of the one of the fundamental solutions – the Airy function Ai(z) – as follows (1.15) The second is a linearly independent solution – a function that is unbounded as z → ∞ and must be dropped. In turn, the solution of Eq. (1.15) has the form
(1.16) Boundary conditions (1.12) and (1.14), on changing to the new variable z, remain unchanged and are now satisfied when z = and z → ∞, respectively. Boundary condition (1.13) becomes the condition (1.17) Corresponding to condition (1.12), taking into account the remarks made on changing to the variable z, we have C1 = 0 for solution (1.16). Substituting solution (1.16) into condition (1.17), we obtain (1.18) Similarly, we have from condition (1.14)
(1.19) It follows from Eqs (1.18) and (1.19) that
(1.20) ¯ we obtain Carrying out Fourier and Laplace transformations of the matching conditions (1.3) and (1.4) for p¯ 1 and A, (1.21) The transform ¯ is found from the transformed equation (1.2) (1.22) The solution of Eq. (1.22) has the form (1.23) ¯ is bounded as y1 = ∞. In the solution obtained we have chosen the minus sign, which ensures that Substituting solution (1.23) into relations (1.21) and eliminating , we obtain (1.24) Eliminating the quantities A¯ and P¯ 1 from Eqs (1.20) and (1.24) and using the Fourier and Laplace transformed expression (1.6), we obtain the dispersion relation
(1.25) Taking the limit in the case of a rigid wall (g → ∞) in dispersion relation (1.25) we get the dispersion relation obtained previously7
when investigating the stability of the boundary layer in the case of transonic flow over a rigid plate. When ␦ = 0, dispersion relation (1.25) gives the dispersion relation obtained previously in Ref. 11 when investigating transonic flow over an elastic surface in the classical three-deck model. The previous analysis of the dispersion relation containing the combination I(), showed (see, for example, Ref. 11), that dispersion relations of this kind have an infinite number of solutions. It can be seen that the inclusion of the elasticity of the surface over which the flow occurs does not increase the number of perturbation waves in the boundary layer compared with the case of flow over a rigid wall (formally due to the fact that the effect of elasticity is described by terms of the dispersion relation (1.25) which depend on g of lower order of smallness in , that already exist in them). The form of dispersion relation (1.25) enables us to express qualitative considerations regarding the effect of the elasticity of the surface over which the flow occurs, based on the fact that a formal consideration of the elasticity leads to the occurrence, in the left-hand side
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A.N. Bogdanov, V.N. Diyesperov / Journal of Applied Mathematics and Mechanics 75 (2011) 357–362
of this dispersion relation, of a term with real and imaginary parts, which depend only on k. Earlier investigations1,6 showed a nonmonotonic behaviour of both and as k increases. Consequently, there are ranges of wave numbers k, for which there will be a value of , corresponding to the chosen value of the wave number k, that is less than when flow takes place over a rigid plate, and this denotes a lower rate of change of the perturbation with time (a smaller increase and a slower damping). Analytically, the roots of dispersion relation (1.25) are easier to find in the plane. Since
then, if we assume k to be a positive √ real number, when Re ˝ > lie below the straight line Re = 3Im.
√
3Im ˝ the roots increase with time Re > 0; consequently, unstable roots
2. The asymptotic of the dispersion relation as k → 0 The search for the relation = (k) can be organized as follows (arising from previous investigations of the stability of interacting boundary layers* :1,6,11 we choose as initial points values of for k = 0. For a zero right-hand side of dispersion relation (1.25) the zeros of its left side occur at the zeros of the derivative of the Airy function, since the integral of the Airy function in the limits from to ∞ takes finite values for any values of . In turn, the derivative of the Airy function only has zeros on the negative part of the real axis.10 There is a denumerable number of such points, and they can be determined with high accuracy by the following formula (m is a natural number)
The asymptotic of the left-hand side of dispersion relation (1.25) can be obtained, for large negative values of , using the asymptotic of the derivative of this function10 for large negative values of the argument. Dropping the first term of this asymptotic, we obtain11
which enables us11 to construct the asymptotic of the roots of dispersion relation (1.25) as k → 0
When k → 0 the zeros of the left-hand side of dispersion relation (1.25) are determined not only by the zeros of the derivative of the Airy function, since there is already one zero: 0 = 0 (it also occurs when ␦ = 0 (Ref. 1). This root is not identical zero: it can be seen from dispersion relation (1.25) that it has no solutions 0 ≡ 0, since for any k it obviously does not satisfy the equality
For the calculated derivative ∂/∂k, for all cases, apart from K∞ = 0, by virtue of dispersion relation (2.7) when k → 0, = 0, we have
Hence, from the start of the change in k from zero, the root 0 departs from the origin of coordinates. A similar result is also obtained in the case when ␦ = 0. However, the asymptotic of the root 0 when 0 → 0 in the cases when ␦ = 0 and ␦ = / 0 is different. For small values of the argument the functions Ai = Ai() we have10
Confining ourselves, in the expansion of the function I(), to the term independent of , we obtain from dispersion relation (1.25) an equation which has two roots. Returning to the variable , we write these roots in the form
(2.1) The first of the roots (2.1) in the principal terms in k is identical with that obtained earlier in Ref. 1. It is stable, since
which, in the case considered, gives a negative value of Re. The second root has a singular pure imaginary term and a sign of the regular part that is opposite to that of the first root; obviously, in the case considered this root increases. When ␦ = 0 the second root drops out and is therefore not described by the classical theory.
∗
Terentev YeD. Non-stationary boundary-layer problems with free interaction. Doctorate Dissertation. Computational Centre, Russian Academy of Sciences, 1986.
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Hence, for small k and → 0, the elasticity of the surface over which the flow occurs leads to damping (growth) of the perturbation, corresponding to the first (second) root, whereas for flow over a solid surface, neutral perturbations correspond to both roots (in this approximation). As k → ∞, the integral of the Airy function mainly defines the asymptotic of the dispersion relation (unlike the asymptotic when k → 0). That is, for dispersion relation (1.25) as k → ∞ the dispersion curves, emerging from the zeros of the arbitrary Airy function, pass through points defined by the zeros of the equation
Their asymptotic representation for large m is as follows:12
We can say once again, that the root of dispersion relation (1.25), which departs to infinity, is determined by the asymptotic ratio of the derivative of the Airy function and the integral of this function, namely (2.2) The asymptotic behaviour of dispersion relation (1.25) as k → ∞ also depends on the asymptotic behaviour of the factor with the square root; it defines the asymptotic of the additional root * as k → ∞. In order to obtain correspondence with the results obtained earlier7 when g → ∞, we also retain in the expansion of dispersion relation (1.25) terms independent of g:
(2.3) When solving relation (2.3), it is much more convenient to convert it into a fourth-degree algebraic equation in : (2.4) It can be seen that when ␦ = 0, the leading term of Eq. (2.4) falls out and the order of the equation is reduced by one. Consequently, when ␦ = / 0 there is an additional root. This root probably corresponds to the wave that arises when there is interaction between the perturbation of the external inviscid flow, propagating downstream with the boundary layer and is special in the sense that it is not described by the classical three-deck model. As can be seen from dispersion relation (1.25), this special root exists independently of the number of terms remaining in expansion (2.2) The case ␦ = 0 was investigated previously and the following high-frequency asymptotic was obtained1 (2.5) Taking the limit of a rigid wall g → ∞ in asymptotic (2.5) is not feasible, since when g → ∞ we have → ∞ and the statement that the roots of the dispersion relation in the case considered and in the case of a rigid surface correspond to one another is impossible. Nevertheless, correspondence should occur since dispersion relation (1.25) when g → ∞ becomes the dispersion relation for the case of flow over a rigid wall.7 Since it is necessary to indicate this correspondence of the roots, below when g → ∞, to fix our ideas, we will assume that g 1 and k/g → 1. Assuming k is a real number, it is more convenient to convert Eq. (2.4) into an equation with real coefficients. When ␦ = 0 it reduces to a third-order equation (2.6) which has three roots; in the principal terms in k when k → ∞ they can be written in the form (2.7) When g → ∞ the asymptotic 1 = k5/3 + 5kKˆ ∞ + . . . , was obtained in Ref. 6, which is almost identical with the first of the roots of (2.7). On the other hand, it can be seen that the term k5/3 in asymptotic (2.5) is not present. Dispersion relation (2.6) has a singular root (2.8) ∗ K∞
−k2 g 2
2 − ␦ k2
− 2␦k2 g −1
if K∞ = ≡ ∗ , for which this root This root, in the case of flow over a rigid wall (g → ∞), was obtained in Ref. 7 (it differs only in the value of K∞ exists). The root obtained corresponds to the solution of a form of travelling wave and exists, as can be seen, for negative values of the ∗ (in the subsonic region of transonic flow). It can be seen that the elasticity of the wall leads to an increase in the transonic parameter K∞ ∗ (residual negative). When the coefficient of elasticity g decreases the parameter K ∗ increases in modulus modulus of the parameter K∞ ∞ (transition from the transonic range of velocities to the subsonic region).
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We can refine the asymptotic of the singular root (2.8):
∗ . which is a generalization to the case K∞ = / K∞ When ␦ = / 0 dispersion relation (2.4) can be solved by Ferrari’s method. Expanding in series in ␦ 1, in the principal terms in k as k → ∞ we have, in the notation of (2.6),
(2.9) Hence, all the roots are pure imaginary. When ␦ = 0 the last root of (2.9) drops out, and in this sense it is singular, like the case of nonstationary transonic aerodynamics, when one root drops out if the modelling of the propagation of weak non-stationary perturbations is carried out using the Lin–Reissner–Tsien equation.12 The remaining roots are regular in ␦. When g → ∞, the roots (2.9) acquire the form corresponding to the case of a rigid surface over which the flow occurs. The effect of the elasticity of the surface reduces to an increase by (k/g)2 of the value of the transonic parameter, for which development of the perturbations in the flow occurs: instead of flow with a subsonic or sonic velocity, weakly supersonic flow occurs (the reduction in g leads to the transition from transonic to supersonic velocity range). References 1. Savenkov IV. The effect of the elasticity of the surface over which flow occurs on the stability of the boundary layer at transonic velocities of the external flow. Zh Vychisl Mat Mat Fiz 2001;41(1):135–40. 2. Kramer MO. Boundary-layer stabilization by distributed damping. J Aeronaut Sci 1957;24(6):459–60. 3. Kramer MO. Boundary-layer stabilization by distributed damping. J Amer Soc Naval Engrs 1960;72:25–33. 4. Carpenter PW, Garrad AD. The hydrodynamic stability of flow over Kramer-type complaint surfaces Pt1. Tollmien–Schlichiting instabilities. J Fluid Mech 1985;155:465–510. 5. Carpenter PW, Garrad AD. The hydrodynamic stability of flow over Kramer-type complaint surfaces. Pt2. Flow-induced surface instabilies. J Fluid Mech 1986;170:199–232. 6. Ryzhov OS, Savenkov IV. The stability of the boundary layer at transonic velocities of the external flow. Zh Prikl Mekh Tekh Fiz 1990;2:65–71. 7. Bogdanov AN, Diyesperov VN. Modeling of non-stationary transonic flow and the stability of the transonic boundary layer. Prikl Mat Mekh 2005;69(3):394–403. 8. Cole,J.D., Cook, L.P. Transonic aerodynamics. Amsterdam: etc.: North Holland, 1986. 9. Bogdanov AN, Diyesperov VN. Tolmien–Schlichting waves in a transonic boundary layer Excitation from the external flow and from the surface. Prikl Mat Mekh 2007;71(2):289–300. 10. Abramowitz M, Stegun IA, editors. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables.. Washington: National Bureau of Standards; 1964. 11. Zhuk VI. Tolmien–Schlichting Waves and Solitons. Moscow: Nauka; 2001. 12. Bogdanov AN. Higher approximations of the transonic expansion in problems of unsteady transonic flows. Prikl Mat Mekh 1997;61(5):798–811.
Translated by R.C.G.