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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

Nonlinear integrodifferential equations describing the propagation of disturbances in a thin layer of viscous liquid with free surface are studied. These equations admit solutions with weak discontinuities, which are located on the characteristics. The possibility of an unbounded increase in the amplitude of the weak discontinuity and the formation of the shock in the process of flow evolution is established. Differential balance laws approximating the integrodifferential model are proposed. These laws are used to perform numerical simulation of wave propagation in a fluid.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

Since the pioneering work of Kapitsa [1], wave motions of liquid films have been a subject of numerous experimental and theoretical studies [2, 3]. Depth-averaged models, in particular, Benney's evolution equation for the film thickness [4], Shkadov's equations [5] and their modifications [6, 7] are often used to describe wave flow regimes of a thin layer of viscous fluid. If the film thickness is not sufficiently small, then the velocity distribution over the depth may differ significantly from the parabolic law. This leads to the necessity to use the equations of the long-wave approximation [8, 9]. In contrast to the theory of Prandtl's boundary layer, the pressure in the fluid is not specified in advance. It should be determined simultaneously with the velocity field and the fluid depth (this is similar to models with self-induced pressure [10, 11]). A generalized method of characteristics for equations with operator coefficients [12, 13] is suitable for theoretical study of wave propagation in a thin layer of viscous liquid in the long-wave approximation. This approach has already been used to study the wave propagation in a supersonic boundary layer within the framework of the model of viscous-inviscid interaction [14].

In this paper, we apply the theory of generalized characteristics for systems of integrodifferential equations describing the wave propagation in a thin layer of viscous liquid with free surface flowing down an inclined plane. Transport equations for the amplitudes of weak discontinuities are derived and the possibility of a nonlinear wave-breaking is established. Numerical calculations based on “multi-layer” approximation of the model confirm the possibility of shock formation.

2. Formulation of the problem

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

We consider the two-dimensional flow of viscous fluid down an inclined plane in the gravity field (Figure 1). The Navier–Stokes equations in dimensionless variables are written as

  • math image(1)

The boundary conditions at the plane inline image and at the free surface inline image are

  • math image(2)

Here, inline image, inline image, inline image, inline image, inline image, inline image are dimensionless components of the velocity, pressure, Cartesian coordinates and time, respectively; inline image, inline image, inline image, inline image, inline image, and inline image are the corresponding dimensional variables. The parameters L0 and H0 determine the characteristic scales on the x and y axes; the quantity U0 is the characteristic velocity. The kinematic viscosity is denoted by ν0, and θ refers to the angle of inclination of the plane to the horizontal. The constants ρ and g0 are the density of the fluid and the gravity acceleration. The surface tension is not taken into account. The shallowness parameter ε, the Reynolds number Re, and the Froude number Fr are defined as

  • display math
image

Figure 1. Schematic diagram of a thin fluid layer down an incline.

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In what follows, we assume that the Reynolds and Froude numbers satisfy the conditions

  • display math(3)

In the long-wave approximation, the parameter ε is considered small. Neglecting terms of order ε2 in Equations (1) and (2) and taking into account (3), we obtain the approximate model

  • math image(4)

where inline image, inline image, and inline image. The dimensionless fluid pressure obeys the hydrostatic law inline image, the constant p0 is the pressure at the free boundary.

In contrast to Benney's equations [15], the model (4) includes a term with the second derivative with respect to y and additional boundary conditions at the bottom and at the free surface. If the Reynolds number is sufficiently large, such that inline image, then the same scaling excludes in Equations (1) terms with the second order derivatives and simplifies dynamic boundary condition in Equations (2).

To study the mathematical properties of Equations (4), it is convenient to rewrite them in semi-Lagrangian coordinates by the change of the variable inline image, where the function Φ is a solution of the Cauchy problem [16]

  • display math

The Lagrangian variable λ varies in the interval [0, 1]; the values inline image and inline image correspond to the bottom inline image and the free surface inline image. In the new variables, the functions inline image and inline image are determined by the integrodifferential system of equations

  • math image(5)

with the boundary conditions

  • display math(6)

These equations are a direct consequence of the model (4) (the derivation is similar to the case of inviscid flow [17]).

Transformation to the semi-Lagrangian coordinates is a reversible change of variables if inline image (in what follows we assume that inline image). Indeed, let the functions inline image and inline image be found. Then, the Eulerian coordinate y, the fluid layer depth inline image, and the velocity component v can be obtained from the formulas

  • display math

The simplest stationary solution of the Equations (4) is

  • display math(7)

where the depth h and the flow discharge inline image are constants. Let us integrate the first equation of system (4) with respect to y under assumption that the velocity profile is given by formula (7) with the unknown functions inline image and inline image. Using other equations of system (4), we obtain the dispersionless Shkadov model [5]

  • display math(8)

This system is hyperbolic and its characteristic velocities are

  • display math

Further, we show that the integrodifferential model (5) is also hyperbolic (with respect to the spatial variable x) in the sense of [12, 13].

3. Generalized characteristics

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

System (5) can be written in the following form

  • display math(9)

where inline image is the vector of unknown quantities, inline image is the result of the action of the operator A on the vector function inline image:

  • display math

and f is the right-hand side of Equations (5):

  • display math

(the superscript T denotes the transposition). For the time being, we assume the term f with higher order derivatives with respect to λ to be known. Then system (9) has the form of the quasilinear equations with operator coefficients. For this type of systems a generalization of the notions of characteristics and hyperbolicity was proposed [12, 13].

According to [12], the characteristic surface inline image of system (9) is defined by the equation inline image, where inline image is the eigenvalue of the problem

  • display math(10)

(inline image is a smooth test function and I is an identical map). The solution of Equation (10) for the functional inline image is sought in the class of locally integrable or generalized functions. The functional inline image acts on the variable λ for fixed values of t and x. Applying the functional inline image to Equation (9), we obtain the following relation on the characteristic:

  • display math(11)

System (9) is generalized hyperbolic if all the eigenvalues inline image are real and the set of relations on the characteristics (11) is equivalent to the Equations (9), i. e., the system of eigenfunctionals inline image is complete (if a test function inline image is smooth and for each α the equality inline image holds, then inline image).

The characteristic properties of integrodifferential Equations (5) (with zero right side) are studied in [13, 18]. Therefore, below we present the eigenfunctionals inline image and formulate (without proof) the hyperbolicity conditions for Equations (5).

In contrast to hyperbolic differential equations, integrodifferential systems may have a continuous spectrum of the characteristic velocities. In this case, any point of the segment inline image, where inline image and inline image, belongs to the characteristic spectrum of problem (10). Let inline image, inline image, then the corresponding eigenfunctionals are

  • math image

For simplicity, the dependence of the functions on t and x is omitted; the integrals are calculated in the sense of the principal value.

If inline image is a root of the equation

  • display math(12)

problem (10) has a nontrivial solution

  • display math(13)

The derivative inline image takes negative values for inline image and is a positive function for inline image. If inline image, inline image; if inline image or inline image, then inline image. From the above-mentioned properties of the function χ, it follows that Equation (12) has two real roots inline image and inline image outside the interval inline image. Note that in particular case inline image and inline image, the Equation (12) gives the characteristic velocities for the classic shallow water equations: inline image.

The hyperbolicity conditions of Equations (5) are formulated in terms of the analytical function inline image, or more precisely, its limiting values inline image from the upper and lower complex half-planes on the interval inline image:

  • display math

Here, inline image; the subscripts 0 and 1 correspond to the values of the functions for inline image and inline image; i is an imaginary unit.

According to [12, 13], for flows with a velocity profile monotonic in depth, the conditions

  • display math(14)

are necessary and sufficient for the hyperbolicity of Equations (5) if the functions inline image and inline image are smooth over the variable λ (the increment in the argument of the functions inline image is calculated when λ is changing from 0 to 1 for fixed values t and x).

4. Propagation of a weak discontinuity

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

If the hyperbolicity conditions (14) hold, the equations of motion (5) can be transformed to the form (11), which includes derivatives only in the directions tangent to the characteristic surfaces. This makes it possible to construct piecewise-smooth solutions with discontinuities in the derivatives in the direction normal to characteristics.

Following [14], we show that a weak discontinuity may exist only on the characteristics. Let inline image be the surface of a weak discontinuity, given by the equation inline image. Calculating the limiting values of expression (9) on Γ, we get

  • display math

Here, inline image is the jump of the function ψ on Γ. Taking into account the continuity of the tangent derivatives to the surface Γ, we have the equality inline image. It follows that the derivative jumps vector inline image is an eigenfunction of the operator A:

  • display math(15)

Applying the eigenfunctionals inline image to the Equation (15) we obtain the relations

  • display math

Since the system of eigenfunctionals inline image is complete, the solution with a weak discontinuity inline image can exist if the value k coincide with one of the characteristic velocities inline image of system (5). It what follows, we assume that inline image and derive the transport equation for the weak discontinuity amplitude.

The eigenfunction inline image of the operator A, obtained from the equations

  • display math

has the form

  • display math(16)

According to Equation (15), the jump of derivatives is proportional to the eigenfunction of the operator A

  • display math(17)

where the coefficient inline image defines the weak discontinuity amplitude.

Let us differentiate the relation on the characteristic (11) with respect to x

  • display math

(hereinafter the superscript “2” in the notation of F2 and k2 is omitted; action of the functional F is determined by (13)). After calculating the difference between limiting values of this expression on Γ we have

  • display math

Taking into account the identity

  • display math

the previous relation can be reduced to the form

  • display math(18)

Here, inline image is the operator of differentiation along the characteristic of slope inline image. The subscript “1” denotes the limiting values of the functions on one side of the surface Γ.

To determine the value inline image we differentiate the characteristic Equation (12) with respect to x, calculate the limiting values on Γ and express the jumps inline image and inline image by using (17). As a result we get

  • display math(19)

Taking into account the identity inline image, and using formulae (13),(16), and (19), we represent the left-hand side of the Equation (18) in the form

  • display math

where

  • math image

Similarly, we transform the right-hand side of Equation (18). By using formulae (13) and (17), we determine the action of the functional inline image

  • math image

and the result of the action F on the function inline image (f is the right side of Equations (5))

  • display math

On the basis of the previous calculations, we collect the terms in (18) with respect to the powers of ξ. This yields the transport equation for the weak discontinuity amplitude

  • display math(20)

where the coefficient C2 has the form

  • math image

Integrating the Equation (20) along the characteristic we obtain the expression for the weak discontinuity amplitude

  • display math

Here, ξ0 is the weak discontinuity amplitude at inline image. Since the coefficient C1 is negative (inline image), an unbounded growth of ξ is possible in the case inline image. The moment inline image of the gradient catastrophe is determined from the equation

  • display math

The integration of (17) with respect to λ from 0 to 1 by using (12) gives inline image. Therefore, a nonlinear breaking occurs for inline image.

5. Approximate differential model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

As we have shown above, the evolution of the smooth solution of the integrodifferential system (5) can involve a gradient catastrophe. The further description of the solution is possible only in the class of discontinuous functions. It leads to the necessity to formulate the model in the form of balance laws. To do this, we will present (5) and (6) in conservative form.

For unknowns

  • display math

the following conservative formulation of the Equations (5) and (6) is used

  • math image(21)

Let us show that the systems (5), (6), and (21) are equivalent for smooth solutions. Obviously, system (21) is a consequence of Equations (5) and (6), obtained by simple transformations. Inversely, integrating the second equation in (21) with respect to λ gives

  • display math(22)

Next, we multiply (22) by H and integrate with respect to λ from 0 to 1 by using the first and the last equations in (21). This yields the equality

  • display math

Comparison of this expression with the third equation in (21) allows one to determine the function inline image and establish the equivalence of the systems under consideration.

Following [17], to derive differential balance laws that approximate the integrodifferential model (21), we divide the segment [0, 1] into M intervals (inline image) and introduce the variables

  • math image(23)

Taking into account the identity inline image and using a piecewise linear approximation for horizontal velocity

  • display math(24)

we have the equalities

  • display math(25)

Next, we integrate the first two equations in (21) with respect to λ over the interval inline image and use formulae (25) and boundary conditions inline image, inline image. Doing so, we obtain for unknown functions

  • display math

the following system of balance laws, consisting of inline image differential equations:

  • math image(26)

where

  • math image

The hyperbolic system of differential balance laws (26), in contrast to the integrodifferential Equations (21), can be solved numerically using standard Godunov type methods. In this case, it is reasonable to use central schemes because system (26) contains a large number of unknown quantities, and exact or approximate solutions of the Riemann problem are difficult to obtain. Below, we present numerical modeling of wave motion of a viscous fluid using multilayered Equations (26) and dispersionless Shkadov model (8).

6. Numerical results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

We implement here the Nessyahu–Tadmor second-order central scheme [19, 20]

  • math image(27)

This scheme approximates the system of balance laws of the form

  • display math

Here, inline image is the spatial grid spacing, while inline image is the time-step satisfying the Courant condition, and inline image. Values inline image and inline image are approximations of the first-order derivatives with respect to x, calculated according to the “ENO limiter” procedure [21]

  • math image

We calculate the solution in the domain inline image up to the time-step inline image, which is chosen in such a way that the perturbations do not reach the boundaries, and one can use initial values as boundary conditions for unknown functions. The calculation domain on the x axis is divided into N cells, the cell centers are denoted by inline image. The Courant number CFL is chosen as 0.475. To find the maximum characteristic velocity inline image one has to solve the equation

  • display math

which is obtained by substituting of the approximation (24) into (12). The root inline image of this equation, in the vicinity of the point inline image, is calculated using Newton's iterative method with the accuracy 10−6. The time step is determined by the formula inline image.

Consider the initial velocity profile in the form (7). The fluid depth h and the discharge inline image are shown in Figures 2(a) and 3(a) (for the first and the second test). The constants a, ν, and g are equal to 0.05, 0.01, and 1, respectively. For the multilayered model (26) at inline image we chose inline image (inline image). Using  (23) we determine corresponding values of inline image and inline image. We perform calculation in the domain inline image for inline image and inline image.

In the first test (Figure 2), a shock is formed during the evolution of the flow with a weak discontinuity. The fluid depths and discharges (especially in the vicinity of the discontinuity), obtained on the base of the models (26) and (8), are essentially different. Furthermore, the model (8) gives a higher speed of propagation of the shock. According to the multilayered model (26), during the flow evolution the velocity distribution over the depth deviates from the parabolic law (7). The largest deviation is observed near the free surface.

image

Figure 2. The growth of a weak discontinuity: (a) the depth h and the discharge q at inline image; (b) the depth h and the discharge q at inline image (curve ML refers to calculation using model (26), curve Shk—model (8)); (c) the velocity profile inline image for inline image at inline image).

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In the second test (Figure 3) the initial data are chosen so that the strong discontinuity does not arise in the evolution of the flow. In this case, an expanding region connecting two stationary Nusselt solutions with different depths is formed. For this initial configuration the calculations using the models (26) and (8) give similar results.

image

Figure 3. The decay of a weak discontinuity: (a) the depth h and the discharge q at inline image; (b) the depth h and the discharge q at inline image (curve ML refers to calculation using model (26), curve Shk—model (8)); (c) the velocity profile inline image for inline image at inline image).

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7. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

The main goal of this work was to show a possibility of a nonlinear wave-breaking in the framework of a long-wave model for viscous flows (4) (or (5),(6)). By extending the theory of characteristics to systems with operator coefficients, it is established that the equations of motion admit solutions with discontinuities of the derivatives. The transport Equation (20) for the weak-discontinuity amplitude along the characteristic surface is derived and the infinite growth of its solution is shown. Thus, the presence in the equations of the second derivative terms with respect to the variable y (or λ) does not improve the solution smoothness with respect to the variable x. The balance laws (26) approximating the original integrodifferential model are proposed. Comparison of numerical results obtained for these balance laws and depth-averaged model (8) shows that the more complex system (26) is needed in the case of shock formation (Figure 2). When solution is smooth, depth-averaged model (8) can be used to perform calculations of wave propagation in a thin layer of viscous fluid (Figure 3).

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References

This work was supported by the Ministry of Education and Science of the Russian Federation (agreement 2012-1.5-8503), Integration Project of SB RAS No. 44, and the Russian Foundation for Basic Research (grant No. 13-01-00249).

The authors thank S. L. Gavrilyuk for fruitful discussions.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of the problem
  5. 3. Generalized characteristics
  6. 4. Propagation of a weak discontinuity
  7. 5. Approximate differential model
  8. 6. Numerical results
  9. 7. Conclusion
  10. Acknowledgments
  11. References
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