Acta Astronautica 67 (2010) 1053–1058
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Propagation of nonlinear travelling waves in Darcy-type porous media M. Singh n, L.P. Singh, Akmal Husain Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi 221005, India
a r t i c l e in fo
Article history: Received 29 January 2010 Received in revised form 17 June 2010 Accepted 22 June 2010 Available online 10 July 2010
A wave front expansion technique is used to analyze the nonlinear wave propagation in one-dimensional, unsteady, compressible ﬂow in a Darcy-type porous media. The analysis leads to an evolution equation for the slope of the wave front. This is an ordinary differential equation, which can be integrated to obtain a closed form solution. The solution may admit a singularity for compression waves. A general formula for the computation of shock formation distance is obtained. The effects of area variation, axial temperature gradient and porosity of the medium on the steepening of compressive wave front into a shock and the shock formation distance are investigated. Two examples highlighting these effects are also presented in the paper. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Compression wave Wave front expansion technique Darcy-type porous media
1. Introduction In recent years, considerable attention has been devoted to the study of ﬂow in porous media due to its applications in astrophysics, geophysics, space science, mining, oil exploration, chemical engineering, etc. Early works in the ﬁeld of wave propagation in porous medium include the book  in which the equation of motion for linear wave propagation in porous media is derived. The problem of linear acoustic wave propagation in porous media has been investigated in . Here, the equation of motion under Forchheimer’s ﬂow law is derived, and a modiﬁcation of Forchheimer–Brinkman equation was proposed in , which is used to study the nonlinear wave propagation in saturated porous media. Also, the wave propagation phenomenon in rigid porous media based on fractional calculus model has been studied in , (see also  and references therein for industrial applications in the area of poroacoustics). The R–H relations for shock wave in very porous medium are
Corresponding author. E-mail address: [email protected]
0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.06.041
examined in . An important contribution developing a theory for the one-dimensional steady compressible ﬂow of gas through a porous plate has been made in . The properties of compressible gas ﬂow in a porous have been carried out in . A list of notable works in the ﬁeld of poroacoustics may be seen in the paper . The problems of nonlinear wave propagation in nonuniform ﬂow have been studied by many authors in various gas dynamic regimes using wave front expansion technique [10–12]. The effect of area and axial temperature gradients on the nonlinear distortion of travelling waves in different gas-dynamics media is analyzed in  and an evolution equation for the slope of the wave front in the closed form is derived. The property of a system, which describes the propagation of small perturbations in porous media is examined in . A problem of propagation of weak disturbances in the combustion of compressible porous fuels is analyzed in . Hsiao and Pan  studied the initial value problem for the system of compressible adiabatic ﬂow through porous media in the one space dimension with ﬁxed boundary condition. Pan  conjectured that Darcy’s law governs the motion of compressible porous media in large time adiabatic ﬂow.
M. Singh et al. / Acta Astronautica 67 (2010) 1053–1058
In the present study, the wave front expansion technique is used to analyze the nonlinear steepening of travelling waves in a ﬂuid that saturates a ﬁxed, rigid, homogeneous and isotropic porous media. The effect of pores in the ﬂow region has been incorporated using the Darcy law given as follows [3,14,15]
system possessing three real characteristics, u+a, u a and u, where a=(gp/r)1/2 is the sound speed . Using thermodynamic relations in Eq. (4), we get @s @s mw 2 þu ¼ T u : ð6Þ @t @x Kr
rP ¼ ðmw=KÞu,
On using thermodynamic relations and Eq. (5), Eqs. (2), (3) and (6) may be reduced to the following form [10–12].
where P is the intrinsic pressure, u is the velocity vector and m, K and w( o1) are the positive constants denoting the dynamic viscosity, permeability and porosity, respectively. Also, the ﬁltration (or Darcy) velocity vector u is related to the intrinsic velocity vector u by the Dupuit– Forchheimer relationship u = wu. Further, it may be noted here that the Darcy’s law does not represent a balance of force . In fact, Darcy law is equivalent to a body force term. The analysis leads to an evolution equation for the slope of the wave front. It is found that the rate of steepening of compressive wave front into a shock is slowed down in porous medium as compared to a perfect gas. The possibility of solution admitting a singularity for compression waves has been analyzed. Effects of area variation and entropy gradient on the shock formation distance in porous media are also studied. In addition, we compare/contrast our ﬁndings with those of the non-porous medium with two examples.
@a @a g1 @u g1 dA þ Au þ Aa þ ua ¼ 0, @t @t 2 @x 2 dx
@u @u 2 @a a2 @s mw g 1=ðg1Þ s=R þu þ þ a e u ¼ 0, @t @x g1 @x gR @x K a2
@s @s mwR g g=ðg1Þ s=R 2 þu ¼ e u : @t @x K a2
In the present work we aim to derive an evolution equation for the ﬁrst derivative of the particle velocity at the wave front in a Darcy-type porous medium. The evolution equation is a nonlinear ordinary differential equation. A general theory on this, usually referred as wave front expansion technique, may be seen in . Here, it is assumed that the wave at the wave front is discontinuous in its ﬁrst order derivative, i.e., dependent variables such as density, velocity and pressure of the gas have a discontinuous ﬁrst order derivatives at the leading edge of the wave.
2. Governing equations
3. Expansion near the wave front
The basic equations for a viscous, compressible, homogeneous ﬂuid ﬂowing homentropically in a ﬁxed and rigid, non-thermally conducting, isotropic porous medium of permeability K and porosity w can be written down in the following familiar form 
For a uniform entropy ﬁeld, the system of Eqs. (2)–(4) can be conveniently analyzed in characteristic co-ordinate system. If x(t)= X(t) be the position of the wave front, x(t)4 X(t) the position of the ﬂuid particle in undisturbed region and x(t)oX(t) the position of the ﬂuid particle in disturbed region, the velocity of the wave front will be
rt þ urx þ rux þ ut þuux þ
ru dA A dx
mw 2 et þ uex 2 ðrt þ urx Þ ¼ u : Kr r p
dx ¼ X_ ðtÞ ¼ aðx,tÞ9xðtÞ ¼ XðtÞ dt
and the two set of characteristics denoted C + and C are
Here r is the gas density, p is the pressure, e is the speciﬁc internal energy, A is the cross-sectional area of the duct, g is the speciﬁc heat ratio, t is the time and x is the spatial co-ordinate. Letter subscripts denote partial differentiation unless stated otherwise. The ﬂow variables in the undisturbed region are assumed to have constant values i.e. r = r0, u = 0 and p = p0, where r0 and p0 are the positive constants The above system of Eqs. (2)–(4) are supplemented with an equation of state p= rRT, where R is the gas constant and T is the absolute temperature. Using the equation of state and thermodynamic relation Tds=dh dp/r, where h is the enthalpy, we get the following relation  p ¼ rg expðs=cv Þ,
where cv is the speciﬁc heat at constant volume and s is the entropy. The system of Eqs. (2)–(4) form a hyperbolic
dx dx ¼ ðu þ aÞ and ¼ ðuaÞ: dt dt
In the present analysis, the frame of reference is ﬁxed to the wave front and wave front is moving with velocity X_ ðtÞ with respect to the ground. Let x be the position of the particle on the wave. Then x = 0, where x = x X(t) denote the position of the wave front. The dependent variables a, u, A and s appearing in Eqs. (7)–(9) are expanded in powers of x about the wave front for x 40 and x o0. If the ﬁrst order derivatives are discontinuous, the appropriate expansions are  9 x2 > aðx,tÞ ¼ a0 ðXðtÞÞ þ xau0 ðXðtÞÞþ a00 0 ðXðtÞÞ þ . . ., > > > 2 > > > > uðx,tÞ ¼ 0, > = 2 for x 40, x 00 Aðx,tÞ ¼ A0 ðXðtÞÞ þ xAu0 ðXðtÞÞ þ A 0 ðXðtÞÞ þ. . ., > > > 2 > > > > > x2 > sðx,tÞ ¼ s0 ðXðtÞÞ þ xsu0 ðXðtÞÞ þ s00 0 ðXðtÞÞ þ. . ., ; 2 ð11Þ
M. Singh et al. / Acta Astronautica 67 (2010) 1053–1058
9 > a2 ðtÞ þ . . ., > > > > 2 > > > 2 > > x > > uðx,tÞ ¼ xu1 ðtÞ þ u2 ðtÞ þ . . ., = 2 for x o 0: 2 > x > Aðx,tÞ ¼ A0 ðXðtÞÞ þ xA1 ðtÞ þ A2 ðtÞ þ . . ., > > > > 2 > > > > > x2 ; sðx,tÞ ¼ s0 ðXðtÞÞ þ xs1 ðtÞ þ s2 ðtÞ þ . . .: > 2 aðx,tÞ ¼ a0 ðXðtÞÞ þ xa1 ðtÞ þ
> @t > > > > @uðx,tÞ > _ _ > ¼ ½u1 ðtÞX ðtÞ þ x½uu1 u2 ðtÞX ðtÞ þ . . ., = @t for x o 0 @Aðx,tÞ > > ¼ ½Au0 ðXðtÞÞA1 ðtÞX_ ðtÞ þ x½Au1 ðtÞA2 ðtÞX_ ðtÞ þ . . ., > > > @t > > > > @sðx,tÞ ; ¼ ½su0 ðXðtÞÞs1 ðtÞX_ ðtÞ þ x½su1 ðtÞs2 ðtÞX_ ðtÞ þ . . ., @t
ð13Þ 9 @aðx,tÞ ¼ a1 ðtÞ þ xa2 ðtÞ þ . . ., > > > @x > > > > @uðx,tÞ > ¼ u1 ðtÞ þ xu2 ðtÞ þ . . ., > = @x for x o 0: @sðx,tÞ > > ¼ s1 ðtÞ þ xs2 ðtÞ þ . . ., > > > @x > > > > @Aðx,tÞ ; ¼ A1 ðtÞ þ xA2 ðtÞ þ . . . @x
Substituting the power series expansions (13) and (14) into Eqs.(7)–(9) and collecting terms of like powers of x, we obtain a set of equations governing the variables of each order as follows x0: g1 u1 ¼ 0, ð15Þ au0 a1 þ 2 u1 þ
s1 ¼ 0,
su0 s1 ¼ 0:
x1 : A1 g1 A1 u1 a0 þ a1 u1 þ au1 a0 a2 þ ðau0 a1 Þa0 2 A0 A0 g1 A1 a0 u2 þ a1 u1 þa0 u1 þ ¼ 0, 2 A0
2 2 2 1 2 a a þ uu1 þ u21 þ a ða s þ 2a0 a1 s1 Þ a0 u2 þ g1 0 2 g1 1 gR 0 2 1=ðg1Þ nr0 w g es0 =R u1 ¼ 0, ð19Þ þ K a0 2 su1 a0 s2 þ u1 s1 ¼ 0,
Since the system of Eqs. (15)–(17) are singular with respect to variables a1, u1 and s1, therefore, it cannot be solved for them. Also, the system of Eqs. (18)–(20) cannot be solved for a2, u2 and s2. Eliminating a1, u1 and s1 from (15)–(17) gives su0 ¼ 2gRau0 =ðg1Þa0 :
Here a0(x) and s0(x) are the known values of sound speed and entropy in undisturbed ﬂow; A(x) is the crosssectional area of the duct; and u0(x) is assumed to be zero in undisturbed medium. For x 40 the coefﬁcients of x0 are known and other coefﬁcients will be determined in foregoing analysis. Also, all the derivatives of A0(x) are known on both sides of the wave front, i.e. 00 A1 ðtÞ ¼ Au0 ðXðtÞÞ, A2 ðtÞ ¼ A0 ðXðtÞÞ, etc. Required derivatives of the dependent variables for x o0 may be written as 9 @aðx,tÞ ¼ ½au0 ðXðtÞÞa1 ðtÞX_ ðtÞ þ x½au0 a2 ðtÞX_ ðtÞ þ . . ., > >
where prime denotes differentiation with respect to time.
Similarly, eliminating a2, u2 and s2 from Eqs. (18)–(20) gives the following ﬁrst order nonlinear differential equation (Riccati Eq.) ! 1 a0 ðXðtÞÞAu0 ðXðtÞÞ nwða0 ðXðtÞÞÞ2 þ u1 ðtÞ uu1 ðtÞ þ au0 ðXðtÞÞ þ 2 A0 ðXðtÞÞ Kn 2 Lða0 ð0ÞÞ2
ð22Þ u1 2 ðtÞ ¼ 0: 2 In the above equation, the coefﬁcient of Darcy term has become d = w/Re, where Re = a0(0)lr0/m is the Reynolds number and we deﬁned l= K/L for convenience. We also note that thepcontinuum assumption demandspKﬃﬃﬃnﬃ o0.01, ﬃﬃﬃﬃ where Kn ¼ K =L is the Knudsen number, K is the average molecular free path, L is the physical length scale and l is the characteristic length scales . The ﬁrst term uu1 ðtÞ in Eq. (22) represents the rate of change of slope of the wave front. The second term signiﬁes the change in cross-sectional area and axial variation in the mean temperature, and is linear in u1(t). The third term is nonlinear in u1(t). For a plane wave moving in constant–area duct with homentropic ﬂow, Eq. (22) becomes þ
uu1 ðtÞ þ ðnw=2Kn2 LÞu1 ðtÞ þ ððg þ 1Þ=2Þu21 ðtÞ ¼ 0:
If the nonlinear term in the above equation is neglected we get a solution as u1 = c1e ct, where c ¼ nw=2Kn2 L and c1 is an arbitrary constant, which shows that the slope of the wave front decreases exponentially with time i.e., the wave moves without any nonlinear distortion and decays after a ﬁnite time. However, for a ﬁnite amplitude wave, the slope of the wave front changes in a nonlinear fashion.
4. Nonlinear distortion of the wave front In this section, the analysis of the nonlinear distortion of a wave front moving in a variable area duct with varying temperatures in porous media will be presented. Since the Riccati Eq. (23) is a nonlinear differential equation in u1, therefore its solution cannot be obtained. However, it can be reduced to a linear ﬁrst order ordinary differential equation, if a particular integral is known . Assuming that the position of the wave front, X(t), as an independent variable instead of time and writing du1 du1 dXðtÞ du1 ¼ ¼ a0 ðXðtÞÞ, dt dXðtÞ dt dXðtÞ a0 ðXðtÞÞdu1 1 Au0 ðXðtÞÞa0 ðXðtÞÞ au0 ðXðtÞÞ þ þ 2 AðXðtÞÞ dXðtÞ ! 2 nwða0 ðXðtÞÞÞ g þ1 2 u1 ¼ 0: þ u1 þ 2 Kn2 Lða0 ð0ÞÞ2
M. Singh et al. / Acta Astronautica 67 (2010) 1053–1058
The solution of Eq. (25) is given by Z 1 1 IFð0Þ g þ1 y IFðyÞ þ ¼ dy, u1 ðyÞ u1 ð0Þ IFðyÞ 2IFðyÞ 0 a0 ðyÞ
compression wave front having slope given by (32) will steepen into shock. ð26Þ 6. Compression wave travelling in a homentropic medium
where the integrating factor is " Z ! # 1 Au0 ðyÞ au0 ðyÞ nwa0 ðXðtÞÞ þ IFðyÞ ¼ exp dy , 2 A0 ðyÞ a0 ðyÞ Kn2 Lða0 ð0ÞÞ2 ð27Þ y= X(t) is the location of the wave front and u1 is the initial slope of the wave front.
5. Compression waves In this section, the nonlinear distortion of compression waves will be analyzed. For a compression wave front, the initial slope u1(0) is negative and may be written as u1 ð0Þ ¼ 9u1 ð0Þ9:
which gives the slope of the compression wave front at a position y. Assume that the right hand side of Eq. (29) is zero at some ﬁnite values of y= ys; this gives the condition for the shock formation 2IFð0Þ Ry : ðg þ 1Þ 0 s ðIFðyÞ=a0 ðyÞdyÞ
ðg þ 1Þ
2IFð0Þ R1 : 0 IFðyÞ=a0 ðyÞdy
2IFð0Þ Ry : ðg þ1Þ 0 s IFðyÞ=a0 ðyÞdy
1 g þ1 9u1 ð0Þ9 2a0 ð0Þ
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! A0 ð0Þ dy : y A0 ðyÞ 2Kn2 Lða0 ð0ÞÞ
ð33Þ The shock formation time t, in the form of shock formation distance, is given as Z ys t¼ dy=a0 ðyÞ: ð34Þ 0
The shock formation distance b for plane wave front in homentropic conditions become Z ys pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ expðnwy=ð2Kn2 La0 ð0ÞÞÞ A0 ð0Þ=A0 ðyÞdy 0
Inequality (31) gives the maximum value of initial slope of a wave front that will not steepen into a shock. Therefore, for a compression wave front that steepens into a shock must satisfy the following condition u1 ð0Þ 4
In view of condition (30) one can infer from Eq. (29) that the ﬁrst derivative of the wave front becomes inﬁnite and a shock is formed. Further, the steepening of a compression wave front into a shock is greatly inﬂuenced by variations in the cross-sectional area of the duct, entropy and porosity of the medium. If for a given wave front the slope given by (29) always remains negative, which shows that the bracket on the right hand side of Ry (29) is positive. Also, 0 IFðyÞ=a0 ðyÞdy is an increasing R1 function of y and has a maximum value 0 IFðyÞ=a0 ðyÞdy, consequently Eq. (29) yields the following condition 9u1 ð0Þ9o
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 nw A0 ðyÞ y ¼ exp 2 u1 ðyÞ A0 ð0Þ 2Kn Lða0 ð0ÞÞ
On using Eq. (28), Eq.(26) reduces to the following form ! Z 1 IFð0Þ 1 g þ 1 y IFðyÞ ¼ dy , ð29Þ u1 ðyÞ IFðyÞ 9u1 ð0Þ9 2IFð0Þ 0 a0 ðyÞ
9u1 ð0Þ9 ¼
In the previous analysis, the effects of change in crosssectional area of the duct and the temperature variation in the duct on the wave form were discussed. In this subsection, the effect of area variation alone in a porous medium on the wave form distortion will be analyzed. Although Eq. (29) is applicable for any smooth duct, in the present discussion only ducts with monotonically increasing or decreasing cross-section will be considered. In a homentropic environment a0(x) is constant. Hence, Eq. (29) reduces
It may be noted here that if the integral on the right hand side of (32) does not converge, all the compression wave fronts will steepen into shock after travelling a ﬁnite distance given by (30). If the integral converges, only the
¼ b ¼ 2a0 ð0Þ=ð9u1 ð0Þ9ðg þ1ÞÞ:
From the above equation it is clear that for a diverging duct the shock formation is delayed as compared to a duct with constant cross-sectional area. Further, Eq. (35) shows that the effect of porosity of the medium is to further delay the shock formation. In converging ducts the shock will form before b. Therefore, for a homentropic environment, the shock condition (32) reduces to 9u1 ð0Þ94
ðg þ 1Þ
R ys 0
2a0 ð0Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , expðDy=2Þ A0 ð0Þ=A0 ðyÞdy
where D ¼ nw=Kn2 La0 ð0Þ, is the damping coefﬁcient. Example (a): Consider an example of a plane compression wave front travelling in a duct, whose cross-sectional area varies as A0(x)= A0(0)(1 + ax)n, where a 40 and n is a real number. The function may represent four different types of ducts depending upon the value of n. Our integral of interest is Z
Z 1 A0 ð0Þ 1=2 expðDy=2Þ expðDy=2Þdy ¼ dy A0 ðyÞ ð1 þ ayÞn=2 0
¼ 1, D o0, a o 0 and n r2 n D , D Z0, a 4 0 and n 4 2: ¼ expðD=2aÞD1 þ n=2 an=2 G 1 , 2 2a
M. Singh et al. / Acta Astronautica 67 (2010) 1053–1058
From Eq. (38), we conclude that a compression wave front travelling in a duct for which D o0, a o0, and n r2 will always steepen into a shock wave. The condition for shock formation will be obtained from Eqs. (33) and (38) and may be written as 9u1 ð0Þ9 4
, ðg þ 1ÞexpðD=2aÞD1 þ n=2 an=2 G 1 n2 , 2Da
Expression of shock formation distance in polynomial area variation becomes h
G 1 ,Dð1 þ ays Þ=2a ¼ G 1 ,D=2a bD1n=2 an=2 eD=2a : ð40Þ
where G is the incomplete Gamma function.
The effect of initial slope on the evolution of the compression wave front in a porous media with polynomial area variation is presented in Fig. 1.
200 0 200 400 600 u1(0) = –200s-1 800
u1(0) = –240s-1 u1(0) = –343s-1
Fig. 1. The evolution of the slope of compression wave front for different values of initial slope, with a polynomial area variation, dotted line—perfect gas and dashed line—porous media.
y Fig. 2. The evolution of the slope of compression wave front for different values of initial slope, dotted line—perfect gas, and dashed line—porous media, with exponential area variation.
M. Singh et al. / Acta Astronautica 67 (2010) 1053–1058
Example (b): In this example, we consider an exponential horn duct A0(x)= A0(0)eax, where a is the ﬂare constant of the horn. The horn is diverging when a 40 and converging when a o0. The value of the integral in Eq. (36) reduces to Z 1 Z 1 A0 ð0Þ 1=2 expðDy=2Þdy ¼ exp½ðD þ aÞ=2ydy A0 ðyÞ 0 0 ð41Þ ¼ 1,
D þ a r 0,
¼ 2=ðD þ aÞ,
D þ a 4 0:
Thus, every compression wave front travelling in a converging horn will steepen into a shock. In a diverging horn the condition for shock will be 9u1 ð0Þ94 ðD þ aÞa0 ð0Þ=ðg þ 1Þ:
Shock formation distance with exponential area variation in a Darcy-type porous medium becomes ys ¼
2 lnð1ðD þ aÞb=2Þ : Dþa
The effect of initial slopes on the evolution of the slope of compression wave front in a porous medium with exponential area variation is presented in Fig. 2. 7. Results and discussion The slope of wave front of nonlinear travelling wave is computed using Eq. (33) with polynomial area variation A0(y)= A0(0)(1 + ay)n, and an exponential area variation A0(y)= A0(0)eay, 0ry r1. The case w = 0 corresponds to non-porous medium. The typical values of physical quantities involved in computation are taken as g =1.4, a =1, n=4, a0(0)=343 m/s, L=1 m, n =0.00001466, Kn = 0.000421 and w = 0.476. The results obtained from the computation of Eq. (33) are plotted in Fig. 1 for polynomial area variation and in Fig. 2 for an exponential area variation of a duct. In both cases, the steepening of compression waves are slowed down in case of porous medium as compared to nonporous case. It may also be noted that the compression waves steepen into a shock only if the magnitude of initial slope of wave front is greater than a critical value. 8. Conclusions In the present paper, a problem of propagation of nonlinear travelling waves in a Darcy-type porous medium is studied. Wave front expansion technique is used to analyze the growth and decay of compression waves in a variable area duct. The analysis leads to an evolution equation for the slope of the wave front. The condition for the steepening of compressive wave front and an expression for shock formation distance are obtained. The analyses show that the rate of steepening of compression
wave into a shock wave with variable area ducts in Darcytype porous media is slowed down as compared to the ideal gas, which is illustrated by two examples. Acknowledgement Authors are grateful to the referee for giving useful suggestions and making certain points more explicit.
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