Thermoelastic stability of closed cylindrical shell in supersonic gas flow

Theoretical & Applied Mechanics Letters 9 (2019) 285-288  

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Letter

Thermoelastic stability of closed cylindrical shell in supersonic gas flow G.Y. Baghdasaryana, M.A. Mikilyanb,*, I.A. Vardanyana, P. Marzoccac  

a

Institute of Mechanics, National Academy of Sciences, Yerevan 0019, Armenia Russian-Armenian University, Institute of Mechanics, National Academy of Sciences, Yerevan 0019, Armenia c Engineering, Aerospace Engineering and Aviation, Sir Lawrence Wackett Centre: Air, Space, Land, Sea School of Engineering, RMIT University, PO Box 71 Bundoora VIC 3083, Australia b

H  I  G  H  L  I  G  H  T  S

•  When the edges are free, a constant thermal field does not affect the value of flowing critical speed. •  Depending on the wave number, critical speed has a minimum point. •  Variability of temperature field has a significant effect on the value of the critical speed (it increases) and this effect increases with the increasing wave number. •  Critical speed as a function on wane number has a minimum point, which depends on variability of temperature field. •  Constant component of thermal field has a significant effect the stability of examined system. •  Positive temperatures narrow the area of stability and reduce the value of the critical speed. Moreover, there is a positive value of a constant temperature field above which the cylindrical shell becomes unstable. •  Negative constant temperature leads to the expansion of stability region and to an increase of the value of critical speed. A  R  T  I  C  L  E      I  N  F  O

A  B  S  T  R  A  C  T

 

 

Article history: Received 4 July 2019 Accepted 29 July 2019 Available online 30 September 2019

   

Keywords: Cylindrical shell Supersonic gas flow Thermal field

In this paper the problem of linear stability of a closed cylindrical shell under the action of both non-uniform temperature field and supersonic gas flow is considered. The stability conditions for the unperturbed state of the aerothermoelastic system are obtained. It is shown that, by the combined action of the temperature field and the ambient supersonic flow, the process of linear stability can be controlled and the temperature field affects significantly the critical flutter speed. ©2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).  

 

  A thin isotropic closed circular cylindrical shell of constant thickness h and radius R, located in a variable temperature field T is considered. The cylindrical coordinate system x, φ, r is chosen, the coordinate lines φ and x coincide with the curvature lines of the middle surface of the shell (x along the generatrix and φ along the arc of the cross-section). A supersonic gas flow is considered on the outer side of the shell with an unperturbed velocity U , directed along the axis x . The stability of the aerothermoelastic system are investigated. The statement of the problem in the case of elongated plates is investigated in Ref. [1]. The study is based on the following well-known assumptions: (a) Kirchhoff hypothesis on non-deformable normals [2];  

 

(b) Piston theory aerodynamics is used to compute the aerodynamic pressure [3] (c) Linear law of temperature variation over the shell thickness [4]: T = T0 (x, φ) + (R − r )Θ (x, φ); (d) Neumann's hypothesis on the absence of shifts from temperature changes [5]. For simplicity and clarity, it is assumed that there is a heat exchange from the front surfaces (r = R ± h/2) of the shell with the surroundings according to Newton-Richman law (the surfaces retain a constant temperature T + and T −), and the side surfaces (x = 0 and x = a ) are thermally insulated. Then the problem of thermal conductivity has the following solution:

 

* Corresponding author. E-mail address: [email protected] (M.A. Mikilyan).

 

T = T0 + (r − R)Θ, T0 =

T+ +T− k (T + − T − ) , Θ= , 2 kh − 2λ

(1)

http://dx.doi.org/10.1016/j.taml.2019.05.001 2095-0349/© 2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

286

G.Y. Baghdasaryan et al. / Theoretical & Applied Mechanics Letters 9 (2019) 285-288

where λ is coefficient of thermal conductivity and k is coefficient of heat transfer with the temperature T measured in degree Celsius (°C). Under the action of a stationary temperature field which is non-uniform over the thickness (Θ ̸= 0), shell buckling occurs with deflection w T (x) and longitudinal displacement u T (x) which results in aerodynamic loading. The indicated buckling state is accepted as unperturbed [6] and its stability is investigated under the action of the temperature field and the pressure of the flowing gas flow. Based on the accepted assumptions, from the basic equations, relations and boundary conditions of the theory of thermoelasticity of thin shells, the governing aerothermoelastic equations are obtained. These are characterized by the axisymmetric state of examined system relations: dw T u 0(r ) = w T (x) , u 0(x) = u T (x) − (r − R) , dx

(2)

according to the Kirchhoff-Love hypothesis. These equations with respect to u T and w T are cast as follows d2 u T µ dw T + = 0, dx 2 R dx [ D

(3)

(4)

according to the theory of thin shells. The internal forces of the unperturbed state are described as: ∫

R+h/2

σ011 dr =

R−h/2

[ ] ( ) Eh du T wT + µ − α 1 + µ T , 0 1 − µ2 dx R

∫

R+h/2 0 T22 =

σ022 dr =

R−h/2

[

]

Eh ∂u T w T µ + − α(1 + µ)T0 . 1 − µ2 ∂x R

( ) d2 w T + α 1 + µ Θ = 0 at x = 0, x = a, 2 dx

( ) du T − α 1 + µ T0 = 0 at x = 0, x = a . dx

(5)

(6) (7)

● the edges are hinged and immobile w T = 0,

( ) d2 w T + α 1 + µ Θ = 0 at x = 0, x = a, dx 2

u T = 0 at x = 0, x = a.

µ R

∫x w T (ξ)dξ + δ 0

µx Ra

( ) + (1 − δ)α 1 + µ T0 x,

∫a w T (x)dx 0

(10)

where δ=

{ 0, when the edges are free, 1, when the edges are fixed.

By virtue of Eq. (10) from Eq. (5) one can have: 

 ∫a E µ h w T (x) E hα   T =δ dx − T0 , 1 − µ2 a R 1−µ 0 [ wT 0 T22 =E h − (1 − δ)αT0 R   ] ∫a 2 µ 1 αT 0  +δ  w (x)dx − . T 1 − µ2 Ra 1−µ 0 11

(11)

The problem of determining w T by virtue of Eqs. (5) and (9) reduces to solving the equation D

[ d4 w

T

dx 4

+

12 ( 1 − µ2 δµ2 wT + 2 Rh R Ra

+ (1 − δ)αµ(1 + µ)T0

according to the generalized Hooke's law. −1 Here M = Ua ∞ is the Mach number for the unperturbed flow, a ∞ = æp ∞ ρ −1 is the speed of sound for the unperturbed gas, p ∞ ∞ and ρ ∞ are the pressure and density of the gas in the unperturbed state, æ is the polytropic coefficient, D = E h 3 /12(1 − µ2 ), E is the elastic modulus, µ is the Poisson coefficient, while α is the coefficient of linear thermal expansion of the shell material. The solutions of Eqs. (4) and (5) must satisfy the boundary condition at the shell fixed edges x = 0 and x = a . The following boundary conditions are considered: ● the edges are hinge supported and can freely move along the axis x w T = 0,

uT = −

0

( )] d4 w T 12 du T w T dw T + µ + + æp ∞ M = 0, dx 4 Rh 2 dx R dx

0 T11 =

The solution u T of Eq. (4), satisfying condition Eq. (8), has the following form with the condition w T = 0 on the ends of the shell is also used:

(8) (9)

)]

∫a w T (x)dx 0

(12)

dw T + æp ∞ M = 0, dx

under the boundary conditions either Eq. (6) or (8). The solution of equation (12) that satisfies the specified conditions is obtained. It is not given here due to its bulkiness. On the basis of assumptions (a)–(d) for the undisturbed state, similarly to Ref. [7], the following linear differential system of aerothermoelastic stability equations is obtained: ∂2 u 1 − µ ∂2 u 1 + µ ∂2 v + + ∂x 2 2 ∂φ2 2 ∂x∂φ 0 µ ∂w (1 − µ2 )T22 ∂w + − = 0, R ∂x E Rh ∂x ∂2 v 1 − µ ∂2 v 1 + µ ∂2 u + + ∂φ2 2 ∂x 2 2 ∂x∂φ 1 ∂w h2 ∂ ( w) + − ∆w + 2 = 0, R ∂φ 12R ∂φ R [ µ ∂2 w 1 ∂2 w 2 D ∆ w+ 2 + R ∂x 2 R 2 ∂φ2 ( )] 12 ∂v ∂u w ∂2 w + + µ + + ρ0h 2 2 Rh ∂φ ∂x R ∂t ( 2 ) ( ) 2 ∂w w æp ∞ ∂w 0 ∂ w 0 − T11 2 − T22 + + ρ hε + 0 ∂x ∂φ2 R 2 a ∞ ∂t ∂w æ (æ + 1) 2 dw T ∂w + æp ∞ M + p∞ M = 0, (13) ∂x 2 dx ∂x ( ) ( ) ( ) where u x, φ, t , v x, φ, t , and w x, φ, t are the perturbations of displacements of the points of the middle surface of the shell, ρ 0 is the density of the shell material, and where ε is the coefficient

of linear attenuation.

G.Y. Baghdasaryan et al. / Theoretical & Applied Mechanics Letters 9 (2019) 285-288

When solving specific problems of stability, the boundary conditions for disturbances coming from the edges fixing conditions are added to system Eq. (13). For example, if the edges of the shell are hinged and fixed, then the boundary conditions are represented as: du ∂2 w = 0, v = 0, w = 0, = 0 at x = 0 and x = a. dx ∂x 2

(14)

tion νcr (n) has a minimum point, which depends on Θ . Calculation are performed using α=2.38×10−5 °C−1, k=1200 W/(m2·°C), λ =210 W/(m·°C), μ=0.34, a =1 m, h/a =1/100, and R/a=2. Let us consider the case of axisymmetric problem, when the edges are fixed in the axial direction. In this case, having excluded the function u , the following linear differential equation is obtained from Eq. (13): 

 a 4 2 2 ∫ ∂ w 12 w (x, t )  1 − µ w − µ dw T ∂w + µ D 4 + dx ∂x Rh 2 R R dx ∂x a R

The solution of system Eq. (13), satisfying conditions Eq. (14) is presented in the form (∞ ) ∑ ( ) u x, φ, t = u i (t ) cos λi x sin nθ, i =1

( ) v x, φ, t =

(∞ ∑

( ) w x, φ, t =

0

+

) v i (t ) sin λi x cos nθ,

i =1

(∞ ∑

)

µ a

∫a 0

+ ρh

w i (t ) sin λi x sin nθ,

i =1

λi =

iπ , i = 1, 2, ..., a

+

(15)

where u i (t ), v i (t ), and w i (t ) are the functions of time t to be determined. Substituting Eq. (15) into the system Eq. (13) , applying the orthogonalization process and having restricted us with twoterm approximation, the unknowns u i (t ), v i (t ) are presented using w i (t ). Substituting the obtained expressions into the third equation of the system (13) , the system of ordinary differential equations is received with respect to unknown functions w i (t ). Let us present the solutions of this system in the form w i = y i eλt , from the condition of existence of nontrivial solution the characteristic equation is obtained with respect to λ. Using the Hurwitz theorem, in the case of two-term approximation, the instability regions and critical velocity values are obtained. Numerical analysis is performed and results are depicted in Figs. 1 and 2. These figures show the dependence of νcr on the wave number n in a circumferential direction for several values of components of temperature field. Figure 1 shows that a) when the edges are free, a constant thermal field does not affect the value of νcr and b) depending on the wave number, νcr has a minimum point. Figure 2 shows that (a) variability of temperature field has a significant effect on the value of the critical speed (νcr increases) and this effect increases with the increasing n , and (b) the func-



[ ( 2 )] dw T ∂w  ∂2 w ∂ w d2 w T dx − N1T + N + 1 dx ∂x ∂x 2 ∂x 2 dx 2

( ) ∂2 w æp ∞ ∂w ∂w + ρhε + + æp ∞ M ∂t 2 a ∞ ∂t ∂x

æ (æ + 1) 2

p∞ M 2

dw T ∂w = 0, dx ∂x

(16)

which solution should satisfy the simply supporting conditions w = 0,

∂2 w = 0 at x = 0 and x = a. ∂x 2

Eh αT0 , 1−µ  a  ∫ ∫a Eh w (x, t ) dw T ∂w   ) µ N1 = ( dx + dx . R dx ∂x a 1 − µ2 N1T = −

0

The formulated boundary value problem Eqs. (16) and (17) is solved by the Fourier method and the stability problem of the considered aerothermoelastic system is reduced to the investigation of the sign of real parts of the roots of characteristic equation, depending on the ambient speed and on the characteristics of the temperature field. As a result, the stability conditions are obtained and investigated numerically. On this basis, in Fig. 3, a stability region is constructed on the plane (ν, T0 ) - the case of a uniform temperature field (Θ = 0). T0 = 0

3.5

vcr

vcr

0.9

T0 > 0

0.8

2.0 1.0

0.6

0.5 15

n

20

25

30

Fig. 1.   Dependence of the critical speed ºcr on the wave number n in the case of constant thermal field

Θ=0

1.5

0.7

10

Θ>0

2.5

T0 = 0

1.0

 

3.0

T0 < 0

1.1

(18)

0

 

1.2

(17)

Let us note that in the examined case, w T is the solution of the problem Eqs. (16) and (17). The following notations are from Eq. (16) are used:

Θ=0

 

287

Θ<0

10

 

15

20

25 n

30

35

40

Fig. 2.   Dependence of the critical speed ºcr on the wave number n in the case of variable along the thickness thermal field

288  

G.Y. Baghdasaryan et al. / Theoretical & Applied Mechanics Letters 9 (2019) 285-288 10

Table 1   The value of the critical velocity ºcr at different values of temperature (in °C)

0 −10 −20 −30 −40 −50

 

0

0.01

0.02

0.03

0.04

0.05

0.06

Fig. 3.   Stability region on the plane (º; T0)

Figure 3 shows, that constant component T0 of temperature field has a significant effect the stability of examined system. Namely, (a) if T0 > 0 then the temperature narrows the area of stability and reduces the value of the critical speed. Moreover, there is a positive value of a constant temperature field above which the cylindrical shell becomes unstable, (b) negative constant temperature leads to the expansion of stability region and to an increase of the value of critical speed. The values of the critical speed of ambient supersonic gas are calculated for both homogeneous and variable temperature fields and are listed in Table 1. The influence of a variable temperature field on the stability regions of cylindrical shell is also investigated. The numerical simulation shows that variable temperature field expands the region of instability for positive values and narrows it with negative values of the parameter Θ . Acknowledgements The work was supported by the State Committee on Science and Education of the Ministry of Education and Science of the Republic of Armenia within the framework of the research project (No. SCS 18T-2C149).

T0 ̸= 0, Θ = 0

νcr

T0 = 0, Θ ̸= 0

–100

0.106

–1000

0.012

–50

0.059

–100

0.01209

–30

0.0402

–50

0.0121

–20

0.0309

–10

0.012106

–10

0.0215

10

0.01208

10

0.0027

50

0.012111

20

0.0067

100

0.012115

50

0.0348

500

0.012147

100

0.0818

1000

0.012188

νcr

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[2]

[3] [4]

[5] [6]

[7]

moelastic stability of panels in supersonic flows, J. Therm. Stress. 36 (2013) 915–946. V.Z. Vlasov, The general theory of shells and its applications in engineering, Gostekhteorizdat Publ. Moscow (1949). (in Russian) H. Ashley, C. Zartarian, Piston theory - a new aerodynamic tool for the aeroelastician, J. Aeronaut. Sci. 23 (1956) 1109–1118. V.V. Bolotin, Y.N. Novichkov, Buckling and flutter steady thermally compressed panels are in a supersonic flow, Eng. Journal 1 (1961) 82–96. V. Novatsky, Problems of Thermoelasticity, Publishing House of the Academy of Sciences of the USSR: Moscow (1962) G.E. Baghdasaryan, M.A. Mikilyan, R.O. Sagoyan, Thermoelastic stability of an elongated rectangular plate in a supersonic gas flow, J. Izv. NAS RA Mech. 64 (2011) 51–67. G.E. Baghdasaryan, M.A. Mikilyan, Effects of Magnetoelastic Interactions in Electroconductive Plates and Shells, Springer (2016).