Transcritical Flow Over a Hole

Transcritical flow over a hole R. H. J. Grimshaw 1 , D.-H. Zhang 2 and K. W. Chow 2 Department of Mathematical Sciences, Loughborough University, UK Department of Mechanical Engineering, University of Hong Kong, Hong Kong 1

2

Abstract Transcritical flow over a localised obstacle generates upstream and downstream nonlinear wavetrains. In the weakly nonlinear long-wave regime, this flow has been modeled with the forced Korteweg-de Vries equation, where numerical simulations and asymptotic solutions have demonstrated that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle. Further, it has been shown that when the obstacle is replaced by a step of semi-infinite length, it is found that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore. This result suggests that for flow over a hole, that is a step down followed by a step up, the two wavetrains generated will interact over the hole. In this paper, this situation is explored by numerical simulations of the forced Korteweg-de Vries equation.

1

Introduction

Fluid flow over an obstacle is a long-standing fundamental problem in fluid mechanics. Here we are concerned with the upstream and downstream wavetrains that are generated. A familiar scenario is free-surface flow over a bottom obstacle, when the wavetrains are surface water waves, but the same features arise in many other physical systems. When the flow is not critical, that is, the imposed flow speed U is not close to any linear long wave speed c , linear theory may be used to describe the wave field, (see, for instance, Lighthill (1978) and Whitham (1974)). In this case, stationary lee waves are found downstream for subcritical flow ( U < c ) together with transients propagating both upstream and downstream, while only downstream-propagating transients are found in supercritical flow ( U > c ). However, these linear solutions fail near criticality ( U = c ), as then the wave energy is trapped near the obstacle. In this case it is necessary to invoke weak nonlinearity to obtain a suitable theory, and it is now well known that the forced Korteweg-de Vries (fKdV) equation is an appropriate model. For water waves on an undisturbed depth h , the fKdV equation in non-dimensional form, is 1 1 3 (1) −At − ∆Ax + AAx + Axxx + Fx = 0 . 2 6 2 This is based on the velocity and length scales c, h where c = (gh)1/2 is the linear long-wave speed. A(x, t) is then the dimensionless wave amplitude, that is the surface elevation above the undisturbed depth, while F (x) is the dimensionless obstacle profile, and ∆ = F r − 1 is the criticality parameter, where F r = U/c is the Froude number; ∆ < 0(> 0) defines the subcritical (supercritical) regime. Equation (1) holds in the asymptotic regime when F ∼ 4 , A ∼ 2 , ∂/∂x ∼ , ∂/∂t ∼ 3 , ∆ ∼ 2 , where  0 is the height of the step, L is the separation between the front and the rear steps, and 1/γ measures the width of each step. Note that this form of forcing enables the examination of both a positive step at x = 0 and a negative step at x = L in the same simulation, at least until the time it takes for a disturbance to travel a distance L across the step. Grimshaw et al (2007) considered only positive obstacles, but an immediate inference from their results (an example at exact criticality ∆ = 0 is shown in figure 1) is that for the case of a flow over a negative obstacle ( FM < 0 in (2)), a downstream-propagating wavetrain will be generated at the front negative step, and an upstream-propagating wavetrain will be generated at the rear positive step. These wavetrains will then interact over the obstacle, producing a rather complicated transient state. It is this scenario which we will investigate in this paper, mainly through numerical simulations of the fKdV equation (1), supplemented by some numerical simulations of the full Euler equations. In both cases the initial condition is that A(x, t = 0) = 0 . In section 2 we present a brief summary of the asymptotic theory of Grimshaw et al (2007). Then in section 3 we will describe our numerical simulations of the fKdV equation (1), and we conclude with discussion in section 4

2

2

A 1 0

300

250

200

150 t

100

50

-200

-100

0

100

200

0 300

X

Figure 1: Numerical simulation of the fKdV equation (1) for the positive forcing (2), with FM = 0.1, γ = 0.25, L = 50, ∆ = 0.0 .

3

2 2.1

Asymptotic solutions of the forced Korteweg-de Vries equation Local steady-state solutions

We will suppose that the obstacle is either a positive or a negative step, given by F (x) = 0 ,

for x < 0 ,

F (x) = FM ,

for x > W ,

(3)

and F (x) varies monotonically in 0 < x < W . A positive (negative) step has FM > 0(< 0) . Strictly F (x) should return to zero for some L >> W , as in the forcing function specified by (2), which is that used in all our numerical simulations. But in this section we ignore this, and in effect assume that L → ∞ . In practice this means that the solutions constructed below are only valid for some limited time, depending on how long it takes for a disturbance to travel the distance L . Grimshaw et al (2007) adapted the approach used by Grimshaw & Smyth (1986) for a localized obstacle. Thus, the first step is the construction of a local steady-state solution, using the hydraulic limit in which the dispersive term in (1) is temporarily omitted. Thus in the forcing region we suppose that A = A(x), 0 < x < W , while otherwise A = A−

for x → −∞ ,

(4)

A = A+

for x → ∞ .

(5)

Omitting the dispersive term in (1) it is readily found that −4∆A + 3A2 + 2F = C , or

(6)

3A = 2∆ ± (4∆2 + 3C − 6F )1/2

(7)

There are thus two branches. Application of the limits (4, 5) yield C = −4∆A− + 3A2− = −4∆A+ + 3A2+ + 2FM , giving a connection between A− and A+ . The system is closed by determining the constant C from the long-time limit of the unsteady hydraulic solution, as in Grimshaw & Smyth (1986) and Grimshaw et al (2007). That is, we omit the linear dispersive term in (1) and solve the resulting nonlinear hyperbolic equation, with the initial condition that A(x, t = 0) = 0 . The details are given by Grimshaw et al (2007), and we just summarize the outcome here. Suppose first that the step is positive, FM > 0 , so that ∆≤0: 1/2

0 < ∆ < (3FM /2)

1/2

∆ > (3FM /2)

: :

C = 2FM , 3A− = 2∆ + (4∆2 + 6FM )1/2 , 3A+ = 0 , 2

1/2

C = 2FM − 4∆ /3, 3A− = 2∆ + (6FM ) 2

(8)

, 3A+ = 2∆, (9)

C = 0, 3A− = 0, 3A+ = 2∆ − (4∆ − 6FM )1/2 .

(10)

In all cases, the upstream solution A− > 0 is a “shock” in the hydraulic limit (although in (10) the shock has zero strength and so can be ignored). Such a discontinuous solution needs to be resolved by restoring the dispersive terms in equation (1). The outcome is that the “shock” is replaced with an “undular bore” , that is a modulated periodic wavetrain which resolves the jump. The Whitham modulation theory, see Gurevich & Pitaevskii (1974), Whitham (1974), can then be used to provide an asymptotic description of this wavetrain This process is summarized in the next subsection, or see Grimshaw et al (2007) 4

for further details. Downstream, for ∆ < 0, A+ = 0 and so no resolution is needed. For 0 < ∆ < (3FM /2)1/2 the hydraulic solution with 3A+ = 2∆ > 0 is terminated by a rarefraction wave, but no undular bore solution is needed. For ∆ > (3FM /2)1/2 there is no upstream disturbance, and A+ > 0 so that a rarefraction wave is again needed. Next consider the negative step, FM < 0 , for which the outcome is, ∆≥0: 1/2

−(3|FM |/2)

0 . Here H(x) is the Heaviside function (i.e. H(x) = 1 if x > 0 and H(x) = 0 if x < 0 ). Then the asymptotic method developed of Gurevich & Pitaevskii (1974) and Whitham (1974) seeks a solution in the form of a modulated periodic wave train, A = a{b(m) + cn2 (κ(x − ∆t + V t); m)} + d , 1−m E(m) 4 where b = − , a = mκ2 , m mK(m) 3   3E(m) 3d a 2 − m and V = + − . 2 2 m mK(m)

(16)

(17)

Here cn(x; m) is the Jacobian elliptic function of modulus m and K(m), E(m) are the elliptic integrals of the first and second kind respectively. ( 0 < m < 1 ), a is the wave amplitude, d is the mean level, and V is the wave speed in the negative x direction. This family of solutions contains three free parameters, which are chosen from the set (a, κ, V, d, m) . As m → 1 , cn(x|m) → sech(x) and then the cnoidal wave (16) becomes a solitary wave, riding on a background level d . On the other hand, as m → 0 , cn(x|m) → cos x and so the cnoidal wave (16) collapses to a linear sinusoidal wave (note that in this limit a → 0 ). The asymptotic method of Gurevich & Pitaevskii (1974) and Whitham (1974) is to allow the parameters, that is the amplitude a , the mean level d , the speed V , the wavenumber κ and the modulus m to be slowly varying functions of x and t . The 5

outcome is a set of three nonlinear hyperbolic equations, the so-called Whitham equations, for three of the available free parameters, chosen from the set (a, κ, V, d, m) , or rather better, from appropriate combinations of them. The relevant asymptotic solution corresponding to the initial condition (15) is then constructed in terms of the similarity variable x/t , and is given by   2m(1 − m)K(m) 3A0 x A0 x 1+m− , for − ∆− = < ∆ − < A0 , (18) t 2 E(m) − (1 − m)K(m) 2 t   2E(m) a = 2A0 m , d = A0 m − 1 + . (19) K(m) This asymptotic solution describes an undular bore propagating upstream. Ahead of the wavetrain where x/t < ∆ − A0 , A = 0 and at this end, m → 1 , a → 2A0 and d → 0 ; the leading wave is a solitary wave of amplitude 2A0 relative to a mean level of 0 . Behind the wavetrain where x/t > ∆ + 3A0 /2 , A = A0 and at this end m → 0 , a → 0 , and d → A0 ; the wavetrain is now sinusoidal. Further, it can be shown that on any individual crest in the wavetrain, m → 1 as t → ∞ . In this sense, the undular bore evolves into a train of solitary waves. If A0 < 0 in the initial condition (16), then an “undular bore” solution analogous to that described by (16, 18) does not exist. Instead, the asymptotic solution is an upstreampropagating rarefraction wave, x 0 , there is an upstream jump A− > 0 which is resolved into an undular bore, and a downstream jump A+ < 0 which is resolved by a rarefraction wave. All that is needed is to replace A0 in (18, 19) with A− , and A0 in (20) with A+ . However, because the obstacle occupies a zone around x = 0 , the undular bore must be truncated at x = 0 if necessary, leading to an undular bore attached to the obstacle. This occurs when A− > −2∆/3 and leads to the regime −(FM /2)1/2 < ∆ < (3FM /2)1/2 for an attached undular bore. Otherwise, for ∆ < −(FM /2)1/2 the undular bore is fully formed and is detached (see Grimshaw et al (2007) for more details). For a negative step, FM < 0 , we see from (11, 12, 13) that there is a downstream jump A+ < 0 which is resolved by an undular bore, and an upstream jump A− > 0 which is resolved by a rarefraction wave. The undular bore can now be constructed from ˜ + 3A+ /2 . These (16, 17) and (18, 19) by setting A = A˜ + A+ , A0 = −A+ , ∆ = ∆ ˜ ˜ expression for the undular bore hold for A, ∆ , and the corresponding expressions in terms of A, ∆ are readily constructed. In this case the undular bore is attached when −|6FM |1/2 /2 < ∆ < −|6FM |1/2 /4 , and is fully detached for ∆ > −|6FM |1/2 /4 .

3

Numerical results

The numerical methods used to solve the fKdV equation (1) have been described in Grimshaw et al (2007), and we shall just present the results here. First we consider 6

2

A 1 0

500

400

300

t 200

100

-100

0

100 X

200

0 300

Figure 2: Numerical simulation of the fKdV equation (1) for the forcing (2), with FM = −0.1, γ = 0.25, L = 50, ∆ = 0.0 .

the results for the fKdV equation. In figures 2,4 and 5 we show the simulations of (1) for the forcing (2) with FM = −0.1, γ = 0.25, L = 50 and ∆ = 0.0, 0.2, −0.2 respectively. When ∆ = 0 (figure 2) we see that a downstream undular bore propagates away from the negative step at x = 0 , leaving a depression zone behind, and an upstream undular bore is attached to the positive step at x = 50 , leaving a depression zone behind. This scenario at each step is precisely that to be expected from the analogous simulation for a positive obstacle, as shown in figure 1, and is in complete accord with our theoretical predictions. However, unlike the scenario of figure 1, now these wavetrains interact. In the region between the two steps, and we may assume that the solution could be described as a two-phase modulation solution of the Whitham modulation equations for the KdV equation. This phase is clearer in figure 3, where the step is wider, L = 100 . Eventually the two wavetrains emerge from the interaction, one passing upstream and reaching the step at x = 0 , and the other proceeding downstream and reaching the step at x = 50 . There are then further interaction of each wavetrain with the corresponding step. That with the front step at x = 0 shows evidence that only the largest waves in the wavetrain can pass through the step,and the smaller waves are reflected. However, the interaction with the step at x = 50 seems relatively rather weak, and the wavetrain continues past the step almost unchanged. A supercritical case is shown in figure 4, with ∆ = 0.2 . This has the same scenario as the critical case, but now the downstream-propagating wavetrain from the step at x = 0 is weaker and further downstream; on the other hand the upstream-propagating wavetrain from the step at x = 50 has rather larger waves, but they are more separated, and are not so far upstream. Both these features are in accord with the theory of section 2. The 7

Figure 3: As for figure 2, but with L = 100 .

2

A 1 0

500

400

300

t 200

100

-100

0

100 X

200

Figure 4: As for Figure 2, but with ∆ = 0.2

8

0 300

2

A 1 0

500

400

300

t 200

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-100

0

100 X

200

0 300

Figure 5: As for Figure 2, but with ∆ = −0.2

interaction of these wavetrains is similar to the critical case, but is not as intense. Again, the downstream-propagating wavetrain passes over the rear step at x = 50 without much change, but the interaction of the upstream-propagating wavetrain with the step at x = 0 is again quite strong, with only the leading larger waves passing through the step, and the remaining smaller waves being reflected. Next a subcritical case is shown in figure 5, with ∆ = 0.2 Again there is the same basic scenario as the critical case, but now the upstream-propgating wavetrain from the step at x = 50 is much weaker, and not so far advanced upstream, while the downstreampropagating wavetrain from the step at x = 0 is stronger and is almost attached. Both these features are again in accord with the theory of section 2. The interaction of these wavetrains is similar to the critical case. Again, the downstream-propagating wavetrain passes over the rear step at x = 50 without much change, but the interaction of the upstream-propagating wavetrain with the step at x = 0 is again quite strong, with only the leading larger waves passing through the step, and the remaining smaller waves being reflected.

4

Discussion

In this paper we have complemented our previous study Grimshaw et al (2007) by examining the flow over a long obstacle of negative polarity, that is, a hole. The obstacle consists of a negative step, followed after a long distance, by a positive step. Our numerical simulations of the forced Korteweg-de Vries (fKdV) equation (1) have confirmed that the step down generates a downstream-propagating wavetrain, while the step up generates am upstream-propagating wavetrain. But here these wavetrains interact over the obsta9

L=8 2

A 1 0

600

400

t 200

-100

0

100

0 200

x

Figure 6: Numerical simulation of the Euler equations for the forcing (2), with FM = −0.1, γ = 0.25, L = 50, ∆ = 0.0 .

10

Figure 7: Numerical simulation of the fKdV equation (1) for localized negative forcing, and ∆ = 0.0 .

11

cle, and so the long-time behaviour is quite different from the flow over an obstacle with positive polarity. In our previous paper Grimshaw et al (2007) we successfully compared the numerical solutions of the fKdV equation with analogous simulations of the full Euler equations. However, for the present case of an obstacle with negative polarity, we have found that there are some significant differences, which are the subject of on-going work. In figure 6 we show a preliminary result of a simulation using the full Euler equations for critical flow over a hole, which can be compared with the fKdV simulations of figure 2. In the initial stages, we see that a downstream-propgating wavetrain is generated at x = 0 and an upstream-propagating wavetrain is generated at x = 50 , just as for the fKdV case. But both wavetrains have significantly slower upstream speeds in the Euler equations simulation, and the upstream-propagating wavetrain is also much weaker. The consequent interaction is also much weaker and slower. Indeed we see in figure 6 that at t = 600 the interaction regime remains confined over the step, whereas in figure 2 both wavetrains reach the steps at around t = 300 . This rather unexpected differences between the fKdV simulations and the Euler equations simulation was not seen in the flow over a positive obstacle, reported in Grimshaw et al (2007). A possible explanation is that in the previous case for flow over a positive obstacle the wavetrains propagated upstream from x = 0 into the region x < 0 , and downstream from x = 50 into the region x > 50 . Hence they were in the region where the undisturbed depth and oncoming flow speed were unity (in our non-dimensional coordinates with ∆ = 0 ). But for flow over a negative obstacle as here, the wave trains are in the region 0 < x < 50 , where the undisturbed depth is 1 − FM , and the basic flow speed and the total depth will necessarily have adjusted from unity. Indeed our analysis of section 2.1 implies that at least for small |FM |