The problem of lifting a symmetric convex body from shallow water

European Journal of Mechanics / B Fluids 79 (2020) 297–314

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The problem of lifting a symmetric convex body from shallow water O.A. Kovyrkina a , V.V. Ostapenko a,b , a b

∗

Lavrentyev Institute of Hydrodynamics of SB RAS, Academician M.A. Lavrentiev avenue, 15, Novosibirsk, 630090, Russia Novosibirsk State University, Pirogova Street, 2, Novosibirsk, 630090, Russia

article

info

Article history: Received 30 October 2018 Received in revised form 12 September 2019 Accepted 27 September 2019 Available online 5 October 2019 MSC: 76B15 Keywords: Gravity currents Water exit Shallow water flows

a b s t r a c t A problem of a plane-parallel flow induced by vertical lifting of a symmetric convex body partly immersed in water filling a rectangular prismatic channel with a horizontal bottom is solved within the framework of the shallow water theory. The body width coincides with the channel width, its flat side surfaces are perpendicular to the channel bottom, and its lower downward-convex surface has sufficiently small curvature and is completely immersed in water at the initial time. The liquid flow is obtained analytically in the region adjacent to the lower surface of the body and by means of the numerical solution of shallow water equations by the second-order CABARET (compact accurately boundary-adjusting high-resolution technique) scheme outside this region. Equations that define the motion of the boundary line between the liquid and the lower surface of the body are derived. It is shown that the form of these equations is determined by the sign of the spatial derivative of pressure on this boundary line. Numerical results demonstrating liquid lifting behind the body leaving the water medium are presented. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Lifting of a body from the liquid surface and thereby induced flows are of interest from both theoretical and applied aspects. Mathematical modeling of the lifting process and physical phenomena that should be taken into account are at the stage of preliminary analysis. Weightiness of the liquid, capillary forces and forces of adhesion of liquid particles from the lifted body surface are responsible for liquid separation from the body surface. On the other hand, the action of adhesion forces is localized near the body surface and exerts a minor effect on the global flow of the liquid in the case of body lifting. For comparatively flat bodies with the horizontal size of the wetted part of the body much greater than its immersion depth, a thin layer of the liquid stays on the lifted body surface owing to the forces of adhesion between the liquid and the body surface. However, this thin layer weakly affects the liquid flow in the main domain and the pressure distribution over the wetted part of the body during its lifting. Therefore, the size of the wetted part of the body, which decreases with time, can be approximately determined with adhesion forces being ignored. Theoretical, experimental, and numerical investigations of body lifting from the surface of a deep liquid with the bottom effect being ignored were performed in Reis et al. [1], Piro and ∗ Corresponding author at: Lavrentyev Institute of Hydrodynamics of SB RAS, Academician M.A. Lavrentiev avenue, 15, Novosibirsk, 630090, Russia. E-mail address: [email protected] (V.V. Ostapenko). https://doi.org/10.1016/j.euromechflu.2019.09.020 0997-7546/© 2019 Elsevier Masson SAS. All rights reserved.

Maki [2], Tassin et al. [3], Korobkin [4] and Tassin et al. [5]. Experimental studies of lifting of a circular glass disk from the water surface were performed in Reis et al. [1]. Those studies were aimed at explaining the process of lapping by felines and showed that the process of liquid lifting behind the disk is mainly determined by the gravity and inertia forces, whereas the liquid viscosity and surface tension exert minor effects on the process. The entry into water with a constant negative acceleration and subsequent exit from water were numerically studied for a wedge [2] and a parabolic contour [3]. The time evolution of the wetted surface during the lifting of a plates initially floating at the water surface were experimentally studied by Tassin et al. [5]. Korobkin [4] proposed a linearized model of lifting of a flat body from the free boundary of an infinitely deep liquid. This model is based on the main assumption that the velocity of the boundary of the liquid–body surface contact region is proportional to the local velocity of the flow at the periphery of the contact region. This assumption is similar to that used in Benjamin [6] for modeling the outflow of an ideal incompressible liquid from a rectangular container. The hydrodynamic forces calculated by the model proposed by Korobkin [4] for a wedge and a parabolic contour agree well with the results of the numerical calculations reported in Piro and Maki [2], Tassin et al. [3]. Wave flows arising due to lifting of a rectangular beam partly immersed in shallow water were studied in Kuznetsova and Ostapenko [7], Ostapenko and Kovyrkina [8]. Kuznetsova and Ostapenko [7] considered flows that arise at the first stage of beam lifting from a finite-length channel, when the flat lower surface of the beam is completely immerse in water. Ostapenko and

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Kovyrkina [8] analyzed flows formed when the rectangular beam leaves the water medium filling an infinite-length channel. The present study is a continuation of the investigations Ostapenko and Kovyrkina [8]. Plane-parallel flows arising in the case of vertical lifting of a symmetric convex body partly immersed in shallow water filling an infinite rectangular prismatic channel with a horizontal bottom are considered. Such flows are modeled in the first approximation of the shallow water theory [9,10]; friction, liquid viscosity, and surface tension are ignored. The body width coincides with the channel width, the flat side surfaces of the body are perpendicular to the channel bottom, and the lower downward-convex surface of the body has sufficiently small curvature and is completely immersed in water at the initial time. The liquid flow is obtained analytically in the region adjacent to the lower surface of the body and by means of numerical simulations based on a monotonic modification of the secondorder finite difference CABARET (compact accurately boundaryadjusting high-resolution technique) scheme [11] outside this region. Equations that define the motion of the boundary line between the liquid and the lower surface of the body are derived. It is shown that the form of these equations is determined by the sign of the spatial derivative of pressure on this boundary line. Numerical results demonstrating liquid lifting behind the body leaving the water medium are presented. 2. Formulation of the problem Let us assume that an infinite rectangular prismatic channel with a horizontal bottom is filled by an ideal incompressible liquid with a partly immersed symmetric convex body of length 2L, whose width 2b coincides with the channel width, the flat side surfaces are perpendicular to the channel bottom, and the lower downward-convex surface is completely immersed in water at the initial time (Fig. 1a). At the time t < 0, the liquid and body are at rest. Let us introduce a Cartesian coordinate system whose x axis is aligned on the channel bottom parallel to the side walls at identical distances b from these walls; the z axis is directed vertically upward, normal to the channel bottom. The body is assumed to be located symmetrically with respect to the coordinate plane y0z; therefore, its edges have the coordinates ±L on the x axis. The initial depth of the liquid outside the body, i.e., at |x| > L, is h0 ≪ L, and the pressure on the free surface of the liquid is equal to the atmospheric value, which is assumed to be zero. At the time t < 0, the body occupies the domain

Ω0 = {(x, y, z) : |x| ≤ L, |y| ≤ b, f (x) + h1 ≤ z ≤ z0 }, where z0 > h0 ≥ h2 ≥ h1 > 0,

h2 = f (L) + h1 .

The continuous function f (x) defining the lower surface of the body is even and satisfies the conditions f ′ (x) ≥ 0,

f ′′ (x) ≥ 0 ∀x ∈ (0, L],

f (0) = 0.

At the time t ≥ 0, the body moves vertically upward with a given increasing acceleration H ′′ (t) ≥ 0, where H ′′ (t) ∈ C, and with a given initial velocity H ′ (0) ≥ 0. Assuming that H(0) = h1 , we can uniquely determine the body lifting law H(t), from which it follows that the lower surface of the body at t ≥ 0 is determined by the formula z = f (x) + H(t) for all |x| ≤ L. The wave flows of the liquid arising thereby are induced by the action of the gravity force depending on the gravity-induced acceleration g directed vertically downward. When the body is lifted, the edges of its lower surface leave the water medium at a certain time T1 ≥ 0. The first stage of

the process occurs at T1 > 0, in the time interval (0, T1 ) (Fig. 1b), when the lower surface of the body is still completely immersed in the liquid and the distance HL (t) = f (L)+H(t) from the edges to the channel bottom is smaller than the depth hL (t) = h(L + 0, t) of the liquid adjacent to the body end faces, i.e., HL (t) < hL (t). At this stage, owing to the action of hydrostatic pressure, the depth h(x, t) of the liquid under the body increases during body lifting so that h(x, t) = f (x) + H(t) for all |x| ≤ L. As a result, a liquid flow directed toward the origin is formed; outside the body, this flow consists of depression waves of the initial level h0 . The first stage is finalized when the depth of the liquid adjacent to the end faces of the body becomes smaller than or equal to the height of lifting of the edges of the lower surface of the body, i.e., hL (T1 ) ≤ HL (T1 ). Separation of the body from the liquid occurs at a certain time T2 > T1 . The second stage of the process occurs in the time interval (T1 , T2 ) (Fig. 1c): the edges of the lower surface of the body start to move out of the liquid, while some part of the lower surface still contacts the liquid. We assume that this part of the lower surface of the body is defined by the interval |x| ≤ a(t) whose length a(t) at t ∈ (T1 , T2 ) is a rigorously monotonically decreasing function. At the third stage, at t > T2 , the body becomes separated from water, after which liquid lifting that occurred at the second stage leads to the formation of two diverging waves (Fig. 1d). In Sections 3 and 4, one-dimensional conservation laws of mass and total momentum in the shallow water model are derived from two-dimensional integral conservation laws of mass and horizontal momentum, which describe a plane-parallel flow of an ideal incompressible liquid above a horizontal bottom. It is demonstrated that the mass conservation law for shallow water is obtained without any constraints on two-dimensional flow parameters, and the total momentum conservation law is obtained approximately if two integral inequalities for these parameters are satisfied. Formulas for the flow rate and vertically averaged horizontal velocity of the liquid in the region where the liquid is adjacent to the lower surface of the body are derived in Section 3 from the mass conservation law. A concept of the local hydrostatic approximation is introduced in Section 4. It extends the concept of the long-wave approximation, but is local, in contrast to the latter, and allows one to justify the applicability of shallow water equations in the region where the liquid is adjacent to the body. The local hydrostatic approximation is applied in Section 4 to derive the total momentum conservation law of the shallow water theory and in Section 5 to write the considered problem in dimensionless variables. The liquid flow outside the body at the first stage is studied in Section 6. In Section 7, the equation for the total momentum is used to calculate the pressure between the liquid and the lower surface of the body at the first and second stages of the process. Conditions for propagation of the boundary line x = a(t) of the body–liquid contact with a subcritical velocity are analyzed in Sections 8 and 9. The corresponding situations for critical and supercritical velocities are considered in Sections 10 and 11. Results of numerical simulations are reported in Section 12. 3. Mass conservation law and liquid flow in the domain adjacent to the body Let us fix a space–time domain

Ω = {(x, z , t) : x1 (t) ≤ x ≤ x2 (t), 0 ≤ z ≤ h(x, t), t1 ≤ t ≤ t2 } (1) occupied by the liquid in the channel between the moving vertical planes x = x1 (t) and x = x2 (t) in the time interval [t1 , t2 ]. The mass conservation law for the liquid (1) has the form

∫

x2 x1

⏐ t2 ⏐

h(x, t)⏐⏐ dx + t1

∫

t2 t1

(∫ 0

h(x,t)

)⏐x2 ⏐ v (x, z , t)dz ⏐⏐ dt = 0, x1

(2)

O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

299

Fig. 1. Initial position of the body partly immersed in an undisturbed liquid (a). The first stage of the process in the time interval (0, T1 ): the edges of the body lower surface are in the water medium (b). The second stage of the process in the time interval (T1 , T2 ): the edges of the body lower surface leave the water medium, while some part of the lower surface of the body defined by the interval |x| ≤ a(t) < L still contacts the liquid (c). The third stage of the process at t > T2 : the body is separated from the liquid (d).

where v (x, z , t) is the horizontal velocity of the liquid directed along the x axis. Introducing the notation u(x, t) =

1 h(x, t)

h(x,t)

∫ 0

for the horizontal velocity of the liquid averaged over the depth, we can represent Eq. (2) as the integral law of mass conservation for the shallow water model

∫ v (x, z , t)dz ,

(3)

x2 x1

⏐t2 ⏐

h(x, t)⏐⏐ dx + t1

∫

t2

t1

⏐x2 ⏐

h(x, t)u(x, t)⏐⏐ dt = 0. x1

(4)

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In deriving this conservation law, we did not need any constraints on the parameters of the considered flow. Assuming that the parameters of the flow averaged over the vertical are continuously differentiable, i.e. h(x, t), u(x, t) ∈ C1 , we divide Eq. (4) by (x2 − x1 )(t2 − t1 ) and pass to the limit as x2 → x1 and t2 → t1 . As a result, we obtain the differential form of the mass conservation law for the shallow water model: ht + qx = 0,

q = hu.

(5a, b)

vertical coordinate z. Introducing the notation ˜ v = v − u, we obtain h

∫

v dz = hu + 2

2

h

∫

0

˜ v 2 dz .

0

(14)

For estimating the integral of the pressure p, we assume that the Euler equation for the vertical component of the liquid velocity w (x, z , t) is satisfied in the vertical planes x = x1 (t) and x = x2 (t) bounding domain (1):

At the first two stages, when t ∈ (0, T2 ), the flow domain is divided into two subdomains: the subdomain

dw/dt + pz + g = wt + vwx + wwz + pz + g = 0.

Ω1 (t) = {(x, z , t) : |x| ≤ a(t), 0 ≤ z ≤ h(x, t)} ,

Integrating Eq. (15) twice with respect to the vertical coordinate, first from ξ ∈ [0, h) to h and then from 0 to h, we obtain

(6)

where the liquid is adjacent to the body, and the doubly connected domain

a(t) ∈ [0, L),

p dz =

2

∫ + h˜ p+

h

h

(∫

dt

ξ

0

dw

) dz

dξ .

(16)

If the integrals

with the free upper boundary of the liquid, where t ∈ [0, T1 ];

gh2

0

Ω2 (t) = {(x, z , t) : |x| > a(t), 0 ≤ z ≤ h(x, t)} , a(t) = L,

h

∫

(15)

)⏐x2 ⏐ ˜ v dz ⏐⏐ dt , J1 = 0 t1 x1 ) )⏐x2 ∫ t2 (∫ h (∫ h ⏐ dw J2 = dz dξ ⏐⏐ dt ∫

t ∈ (T1 , T2 ).

If the body lifting law H(t) is known, the liquid depth in subdomain (6) is determined by the formula

t2

(7)

2

dt

ξ

0

t1

h(x, t) = f (x) + H(t).

h

(∫

x1

Substituting this value of the depth into Eq. (5a), we obtain the ordinary differential equation

are equal to zero, then Eq. (13) with allowance for formulas (14) and (16) yields the integral conservation law of the total momentum for the shallow water model

H ′ (t) + qx = 0

∫

(8)

for determining the flow rate of the liquid q = hu in the domain adjacent to the lower surface of the body. Integrating Eq. (8) with respect to x and taking into account the boundary condition q(0, t) = u(0, t) = 0, which follows from the symmetry of the problem around the point x = 0, we find q(x, t) = −

∫

x

H ′ (t)dξ = −xH ′ (t),

|x| ≤ a(t).

(9)

Q (t) = q(a(t), t) = −a(t)H (t),

xH ′ (t) f (x) + H(t)

,

|x| ≤ a(t).

(11)

Therefore, for x = a(t), we obtain U(t) = u(a(t), t) = −

a(t)H ′ (t) f (a(t)) + H(t)

.

(12)

Formulas (7) and (9)–(12) completely determine the liquid flow averaged over the vertical in the region adjacent to the body. 4. Horizontal momentum conservation law In view of formula (3), the horizontal momentum conservation law of the liquid in domain (1) has the form

x1

⏐t2 ∫ ⏐ hu dx⏐⏐ + t1

t2

(

hu2 +

)⏐x2 ∫ t2 ∫ x 2 ⏐ ˜ phx dxdt . + h˜ p ⏐⏐ dt = 2 x1 t1 x

gh2

t1

t1

1

(17) Assuming that the inequalities

⏐∫ ⏐ |J1 | ≪ ⏐⏐

(10)

It follows from formulas (9) and (10) that the flow rate of the liquid in the domain adjacent to the body is independent of the function f (x) defining the shape of the lower surface of the body. Formula (9) for the liquid velocity (3) yields

x2

x1

∫

′

∫

∫

hu dx⏐⏐ +

t2 t1

⏐ )⏐x2 ⏐ ⏐ hu x dt ⏐⏐ , 1

(

2

⏐∫ |J2 | ≤ ⏐⏐ 2

t2

g ⏐

t1

⏐ ⏐ ⏐ ⏐ dt ⏐ ⏐

x2 h2 x 1

(18)

are satisfied, we apply Eqs. (14) and (16) to obtain

0

In particular, for x = a(t), we obtain

u(x, t) = −

⏐t2 ⏐

x2

t2 t1

h

(∫ 0

)⏐x2 ∫ ⏐ ( 2 ) v + p dz ⏐⏐ dt = x1

t2 t1

∫

x2 x1

˜ phx dxdt , (13)

where p = p(x, z , t), ˜ p =˜ p(x, t) = p(x, h(x, t), t) are the specific pressures in the liquid and on the liquid surface, respectively. Following Ostapenko [10], we transform the integrals over the

t2

t2 t1

)⏐x2 ∫ ⏐ v dz ⏐⏐ dt = 2

0

t1

∫

h

(∫

(∫ 0

x1

h

t2 t1

x2 hu2 ⏐x dt + O(J1 ),

(

)⏐

1

(19)

)⏐x2 ∫ t2 ⏐ ⏐x2 g p dz ⏐⏐ dt = h2 ⏐x dt + O(J2 ). 1 2 t1 x 1

In this case, Eq. (13) can be represented in the form of the integral conservation law of the total momentum (17) with accuracy O(J1 + J2 ). The integral conservation law (17) can be obtained without the direct use of inequalities (18) and based on the concept of local hydrostatic approximation [12], which formulation is given below. Definition 1. A plane-parallel liquid flow at a point (x, z , t) locally satisfies the hydrostatic approximation with a small parameter δ ≪ 1 if the following condition is satisfied at this point:

w2 (x, z , t)/C02 ≤ O(δ ) ≪ 1

(20)

√

where w is the vertical velocity of the liquid, C0 = gH0 is the characteristic velocity of propagation of small perturbations, and H0 is the characteristic depth of the flow. Local hydrostatic approximation (20) is a generalization of the classical long-wave approximation in meaning that the characteristic depth H0 of the flow and the characteristic length L0 of surface waves satisfy inequality

ε = H02 /L20 ≪ 1.

(21)

O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

The local hydrostatic approximation (20) allows one to use it in situations where the long-wave approximation (21) is inapplicable. So in the problem considered here the upper boundary of the liquid in domain (6) at the first two stages of the process is adjacent to the lower surface of the body and, therefore, is not free. Its motion (7) is completely determined by the lifting of the body leaving the water medium. This means that the long-wave approximation (21) cannot be used for modeling the liquid flow in domain (6). This approximation also cannot be formally applied in modeling √ hydraulic bores in which vicinity the inequality |hx | ≤ O( ε ) resulting from the condition (21) does not hold. Let us assume that the liquid flow in some neighborhoods Vδ of the side boundaries x = x1 (t) and x = x2 (t) of domain (1) is potential

vz − wx = 0,

(22)

and satisfies the local hydrostatic approximation condition (20). With this in mind we introduce in the domain Vδ the dimensionless variables x∗ =

√ δ H0

v∗ =

v

C0

√

x,

,

z

z∗ = u∗ =

H0 u C0

,

,

t∗ =

δ C0

H0

t,

w w∗ = √ , δ C0

h

h∗ = p∗ =

⏐t2 ∫ x2 ⏐ hu dx⏐⏐ +

x1

t1

( t2 t1

(23)

C0

x1

H∗ =

H h0

,

H∗′ =

LH ′ h0 c0

H′

= √

δ c0

⇒

H∗′′ =

L2 H ′′ h0 c02

t2 ∫ t1

=

H ′′

δg

.

(27)

For the dimensionless variables (27) to be correct, the values of H∗′ and H∗′′ should be bounded from above by some finite values. This means that the velocity and acceleration of body lifting should satisfy the inequalities

√

H ′ /c0 ≤ O( δ ),

H ′′ /g ≤ O(δ ),

(28)

which ensure the validity of the local hydrostatic approximation condition (20) in domain (6) under the body. The input parameters of the considered problem in the dimensionless variables (26) are written as L∗ = (h0 )∗ = g∗ = (c0 )∗ = 1,

f∗ (x∗ ) + (h1 )∗ ≤ 1 ∀ |x∗ | ≤ 1. (29)

. 2

)⏐x2 ∫ ⏐ h2 hu2 + + h˜ p ⏐⏐ dt = 2

As the vertical velocity of the liquid on the upper boundary of domain (6) coincides with the body lifting velocity, equalities (26) yield the following dimensionless variables that define the body lifting law:

,

H0 p

The paper [12] shows that the horizontal momentum conservation law (13) written in the dimensionless variables (23) transforms at δ = 0 to the total momentum conservation law for the shallow water model

∫

301

x2 x1

˜ phx dxdt ,

The shallow water equations obtained in the zero approximation in terms of the parameter δ have the following form in the dimensionless variables (26): ht + qx = 0,

qt + (qu + h2 /2)x = −h˜ px

(30a, b)

where q = hu. Here and everywhere further the asterisk at the dimensionless variables is omitted for brevity. 6. Liquid flow outside the body at the first stage

whose differential form is (hu)t + (hu2 + h2 /2)x = −h˜ px . Returning to the initial dimensional variables by using formulas inverse to formulas (23), we find that inequalities (18) are satisfied. Thus, this derivation of the total momentum conservation law guarantees the fulfillment of inequalities (18) and therefore justifies the derivation of this conservation using formulas (19). As condition (20) and Eq. (22) have to be satisfied only in small neighborhoods of the side boundaries x = x1 (t) and x = x2 (t) of the domain (1), a hydraulic bore can propagate inside this domain, which justifies [12] the applicability of the conservation laws (4) and (17) of the shallow water theory for modeling wave flows with hydraulic bores. 5. Formulation of the considered problem in dimensionless variables In the considered problem of wave flows induced by body lifting from water, the characteristic depth is taken to be the initial depth of the liquid outside the body: H0 = h0

⇒

C0 = c0 =

√

gh0 .

(24)

As the body length is 2L ≫ h0 , the small parameter δ involved in the local hydrostatic approximation condition (20) is defined as

δ = h20 /L2 ≪ 1.

(25)

By virtue of formulas (24) and (25), the dimensionless variables (23) can be represented as x∗ =

w∗ =

x L

, z∗ = Lw

h0 c0

z h0

, t∗ =

, p∗ =

p

. 2

c0

c0 t L

, h∗ =

h h0

, v∗ =

v c0

, (26)

To characterize the flow outside the body at the first stage (in view of the symmetry of the problem and of the fact that ˜ px = 0 at |x| > 1), it is sufficient to solve the initial–boundary-value problem in the domain x > 1, t > 0 for a homogeneous system of the shallow water equations ht + qx = 0,

qt + (qu + h2 /2)x = 0

(31a, b)

with the initial conditions h(x, 0) = 1, q(x, 0) = 0

(32)

and the boundary conditions q(1, t) = Q (t) = −H ′ (t).

(33)

As formulas (9)–(12) remain unchanged being written in the dimensionless variables (26), (27), and (29), the boundary condition (33) follows from formula (10), where a(t) = L = 1. Problem (31)–(33) is independent of the function f (x) defining the lower surface of the body. So a detailed analysis of its solution was performed in Ostapenko and Kovyrkina [8], where lifting of a rectangular beam from shallow water was considered, which corresponds to the simplest case f (x) = 0. Let us discuss the main results that follow from this analysis. For problem (31)–(33) to be well-posed in the case with h2 ≥ hc , where hc = 4/9, it is sufficient to satisfy the condition

(

H ′ (0) < ϕ (h2 ) = 2h2 1 −

√ ) h2

(34)

at the initial time, which means that the first stage of the flow exists; in the case with h2 < hc , it is sufficient to satisfy the condition

|Q | = H ′ ≤ 8/27 ≈ 0.296

(35)

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in the time interval (0, Tc ), where H(Tc ) = hc and Tc ≤ T1 . If these conditions are satisfied, the depth hL (t) of the liquid adjacent to the body end faces is calculated by the formula hL =

1

(

9

( 1 + 2 cos

1 3

( arccos 1 −

27|Q |

)))2

4

≥ hc ,

C =

where the minimum depth hc = 4/9 corresponds to the liquid √ flow with a critical velocity uc = |Q |/hc = hc = 2/3. If the body lifting velocity H ′ is constant and satisfies the wellposedness condition (35) at h2 < hc or condition (34) of the first stage existence at h2 ≥ hc , then the flow in the domain x > 1 at the first stage is a centered s-depression wave [9]. In this case, the final time instant of the first stage T1 = (hL − h2 )/H ′ . 7. Calculation of the pressure between the liquid and the lower surface of the body At the first two stages of the process, the depth, flow rate, and velocity of the liquid in domain (6) under the body are determined by formulas (7), (9), and (11), which remain unchanged being written in the dimensionless variables (26), (27), and (29). Therefore, at |x| ≤ a(t), where t ∈ (0, T2 ), Eq. (30b) can be used to find the pressure ˜ p on the upper boundary of domain (6) adjacent to the lower surface of the body. In view of formulas (7), (9), and (11), Eq. (30b) can be represented as

˜ px =

˜ F (x, t) h(x, t)

=

˜ F (x, t) f (x) + H(t)

,

(36)

xH ′′ + (xf ′ − 2h)x(H ′ )2 /h2 − hf ′ = xH ′′ + (u2 − h)f ′

+2uH ′ .

(37)

Integrating (36) with respect to x, we find

˜ p(x, t) = hL (t) − HL (t) +˜ J(x, t , L),

|x| ≤ L,

(38)

at the first stage and

˜ p(x, t) = ˜ J(x, t , a(t)),

|x| ≤ a(t),

(39)

at the second stage, where the function ˜ J is determined by the formula

˜ J(x, t , a) =

x

∫ a

˜ F (ξ , t)dξ f (ξ ) + H(t)

.

(40)

(

˜ J(x, t , a) =

G(t) a2 − x

) 2 ,

G(t) =

Ha (t) = h(a(t), t) = f (a(t)) + H(t).

(45a, b)

The idea of these definition can be formulated as follows: in the coordinate system moving along the x axis with the velocity a′ (t), the flow on the boundary line is subcritical under condition (42), critical under one of the conditions (43), and supercritical under one of the conditions (44). It was shown [8] that the motion of the boundary line x = a(t) in the case of lifting of a rectangular beam depends on the sign of the pressure ˜ p(x, t), which, in view of formulas (40) and (41), is determined by the sign of the function G(t). In particular, a necessary condition of boundary line motion with the subcritical velocity is satisfaction of the inequality G(t) > 0; if the opposite inequality G(t) ≤ 0 is satisfied, then this line moves with the critical velocity. In the present work, this result is generalized as follows. It is shown that the law of motion of the boundary line x = a(t) depends on the sign of the left one-sided spatial derivative of the pressure Px− (t) = ˜ px (a(t) − 0, t)

(46)

on this line: if the boundary line moves with the subcritical velocity (42), then Px− (t) < 0; if the opposite inequality Px− (t) ≥ 0 is satisfied, then this line moves with the critical velocity a′ = λr .

. (41) H2 the second stage, first of motion of the boundary region.

Definition 2. The boundary line is assumed to propagate with respect to the liquid with a subcritical velocity if the following inequalities are satisfied:

λr = U − C < a′ < λs = U + C ,

(42)

with a critical velocity if one of the following two equalities is satisfied: a′ = λ s ,

(43)

and with a supercritical velocity if one of the following two inequalities is satisfied: a > λs . ′

Let us assume that the boundary line x = a(t) in some time interval (t1 , t2 ) ⊆ (T1 , T2 ) propagates with the subcritical velocity (42). In this case [9], the r-characteristic arriving on the boundary line at t ∈ (t1 , t2 ) brings the value r(t) = u − 2c ,

√

c=

h,

(47a, b)

of the r-invariant from the outer flow region x > a(t). In view of formulas (12) and (45), we obtain the equation

√

U − 2C = −aH ′ /Ha − 2 Ha = r , which yields the implicit formula for the function a(t): aH ′ + ϕ (a) = 0,

( √ ) ϕ (a) = Ha r + 2 Ha ,

Ha = f (a) + H .

In the simplest case of a rectangular beam, where f (a) = 0, formulas (48) yield

2(H ′ )2 − HH ′′

2 For the description of wave flows at all, it is necessary to define the law of line x = a(t) of the body–liquid contact

a < λr ,

Ha (t),

(48a − c)

In the simplest case, where the lifted body is a rectangular beam [see 8], function (40) has the form

′

√

8. Motion of the boundary line with the subcritical velocity

where

( ) ( ) ˜ F (x, t) = − qt + (qu + h2 /2)x = xH ′′ − x2 (H ′ )2 /h + h2 /2 x =

a′ = λ r ,

Here λs and λr are the velocities of the characteristics of system (31), U is the liquid velocity calculated by formula (12), and C is the velocity of propagation of small perturbations determined by the formula

(44)

a(t) = −

H (

√ )

r +2 H .

H′ For calculating the boundary line propagation velocity, we differentiate Eq. (48a) with respect to t. Differentiating the first term in the left-hand side of this equation, we obtain aH ′

(

)′

= a′ H ′ + aH ′′ .

(49)

Differentiating the second term and taking into account (45a) and the formulas C ′ = Ha′ /(2C ),

Ha′ = H ′ + f ′ a′ ,

we find

ϕ ′ = Ha′ (r + 2C ) + Ha (r ′ + 2C ′ ) = (r + 3C )Ha′ + Ha r ′ = λs (H ′ + f ′ a′ ) + Ha r ′ .

(50)

In view of (49) and (50), we obtain H ′ + λs f ′ a′ + λs H ′ + aH ′′ + Ha r ′ = 0.

(

)

(51)

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303

For calculating the derivative r ′ = dr /dt, we use the abbreviated notations

sufficient to satisfy the following inequality in the time interval (t1 , t2 ):

ft+ = ft (a(t), t + 0),

′ h+ x (t) = hx (a(t) + 0, t) < f (a(t)).

fx+ = fx (a(t) + 0, t)

(52)

for one-sided temporal and spatial derivatives. As the r-invariant (47a) at x > a(t) satisfies the equation rt + (u − c)rx = rt + λr rx = 0, its total temporal derivative on the boundary line is r′ =

dr(a(t), t) dt

= rt+ + rx+ a′ = (a′ − λr )rx+ .

(53)

Substituting (53) into (51), we obtain H ′ + λs f ′ + Ha rx+ a′ + λs H ′ + aH ′′ − λr rx+ = 0.

)

(

)

(

⇔

|U | = aH ′ /Ha < C .

(56)

Using Eq. (54) and taking into account inequality (55), we obtain the formula for the boundary line propagation velocity a′ = −

λs H ′ + aH ′′ − λr rx+ . H ′ + λs f ′ + Ha rx+

(57)

It follows from this formula with allowance for inequalities (42) that there is an additional constraint on the body lifting acceleration:

(λr − Ha λs ) rx+ − 2λs H ′ − λ2s f ′ < aH ′′ < (1 − Ha )λr rx+ − 2UH ′ − λr λs f ′ .

(58)

Violation of inequalities (58) means that there are no solutions continuous on the boundary line x = a(t) in the case of body lifting with this acceleration. Such solutions are not considered in the present work. Assuming that the value of a(t1 ) is known, we can apply formulas (57) to uniquely retrieve the position of the boundary line x = a(t) in the time interval (t1 , t2 ). After determining the boundary line position, the flow in the external domain x > a(t) is found by solving the initial–boundary-value problem for system (31) with the initial data formed at t = t1 and the boundary condition h(a(t), t) = Ha (t),

9. Motion of the boundary line with the subcritical velocity at r = const

(55)

As f ′ ≥ 0 and H ′ > 0 at t > 0, then inequality (55) is definitely satisfied under the condition that the value of the brought rinvariant (47a) is constant; therefore, rx+ = 0, and the flow on the boundary line is subcritical:

λs = U + C > 0

It is shown in Appendix A that inequality (61) in the case of motion of the boundary line with the subcritical velocity (42) is equivalent to the inequality Px− (t) < 0, which means that the left one-sided spatial derivative of the surface pressure (46) on the boundary line is negative. It follows from the inequality Px− (t) < 0 that the surface pressure ˜ p is positive in a certain left one-sided neighborhood (a(t) − ε, a(t)) of the boundary line; as a result, the body decelerates lifting of the liquid adjacent to the body in this neighborhood.

(54)

Further we assume that the body is lifted in such a way that H ′ > − λs f ′ + Ha rx+ .

(61)

u(a(t), t) = U(t).

(59)

Let us assume that the value of the r-invariant arrives on the boundary line x = a(t) is constant, i.e., r = const

rx+ = 0

⇒

∀t ∈ (t1 , t2 ).

In this case, Eq. (57) for the boundary line velocity and constrains (58) on the body lifting acceleration take the forms a′ = −

λs H ′ + aH ′′ , H ′ + λs f ′

( ) ( ) − 2λs H ′ + λ2s f ′ < aH ′′ < − 2UH ′ + λr λs f ′ .

(62)

The right inequality of (62) is equivalent to inequality (A.6); therefore, it follows from the condition of local correctness (61). As f ′ ≥ 0, H ′′ ≥ 0, and H ′ > 0 at t > 0, the left inequality of (62) is definitely satisfied under the condition that the flow on the boundary line is subcritical. This means that inequalities (62) do not lead to additional constraints on the law of body lifting under condition (56). We also assume that the r-characteristics arriving on the boundary line at t ∈ (t1 , t2 ) bring the initial value r = r0 = √ −2 h0 = −2 of the r-invariant from the unperturbed part of the external flow x > a(t). In this case, the function (48b) takes the form

ϕ (a) = 2Ha

(√

)

Ha − 1 ,

Ha = f (a) + H .

(63)

As it follows from Eq. (48a) that ϕ (a) < 0 at a > 0, then formulas (63) yield H(t) ≤ Ha (t) < 1 ∀t ∈ (T1 , t2 ).

As only one characteristic arrives at t ∈ (t1 , t2 ) at each point of the boundary line x = a(t) from the interior of the computational domain x > a(t), it is sufficient to set one of the boundary conditions (59) for the problem to be well-posed. At the same time, the function a(t) involved into these conditions is obtained with allowance for the value of the r-invariant brought to the boundary line x = a(t). Therefore, setting two boundary conditions (59) simultaneously does not make the problem ill-posed. For the thus-obtained solution to be correct, it is necessary to satisfy the condition

This means that the level of the liquid in the domain |x| < a(t), where it is adjacent to the body, cannot exceed the initial level of the liquid h0 = 1 outside the body in the case considered here. If the time interval (t1 , t2 ), to which the initial value r0 of the r-invariant is brought, is located at the end of the second stage, i.e. t2 = T2 , then separation of the body from the liquid occurs at the origin of the coordinate system x = 0, at the zero length of the contact region a(T2 ) = 0, at the depth

h(x, t) < f (x) + H(t)

coinciding with the initial depth of the liquid outside the body. In this case, separation of the body from the liquid occurs at the time T2 = (h0 − h1 )/H ′ if the √ body is lifted with a constant velocity H ′ and at the time T2 = 2(h0 − h1 )/H ′′ if the body moves with a constant acceleration H ′′ and zero initial velocity.

∀x ∈ (a(t), 1) ∀t ∈ (t1 , t2 ).

(60)

Its validity in the entire interval (a(t), 1) is verified numerically. At the same time, for condition (60) to be valid in a certain right one-sided neighborhood (a(t), a(t) + ε ) of the boundary line, it is

Ha (T2 ) = f (0) + H(T2 ) = H(T2 ) = 1,

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O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

10. On motion of the boundary line with the supercritical or critical velocity Let us assume that the boundary line x = a(t) in some time interval (t1 , t2 ) ⊆ (T1 , T2 ) propagates with the supercritical velocity a′ > λs = U + C or with the critical velocity a′ = λs and simultaneously either coincides with one of the s-characteristics of system (31) or is an envelope of the family of these s-characteristics arriving from the domain x > a(t). In this case, the r- and s-characteristics arriving on the boundary line at t ∈ (t1 , t2 ) bring the values r(t) = u − 2c ,

s(t) = u + 2c ,

√

c=

h,

(64)

of the r- and s-invariants from the outer flow region x > a(t). On the boundary line formulas (64) yield s+r

√

(s − r)2

s−r

, C = Ha = ⇒ Ha = f (a)+H = 2 4 16 In view of formula (12) and of the fact that s > r, we have U =

UHa

(s + r)(s − r)2

.

(s2 − r 2 )(s − r)

=− =− < 0. a 32a 32a This means that the boundary line can move in such a way only when the body enters the liquid, not when it is lifted from the liquid. Let us assume that the boundary line x = a(t) in the time interval (t1 , t2 ) ⊆ (T1 , T2 ) propagates with the supercritical velocity ′

H =−

a′ < λr = U − C

(65)

or with the critical velocity a = λr and simultaneously is an envelope of the family of r-characteristics moving away from this line to the domain x > a(t). In this case, the flow in the interval |x| < a(t), where the liquid is adjacent to the body, is independent of the external flow at |x| > a(t); therefore, the boundary line motion cannot be determined by formula (12) within the framework of the shallow water theory. If the boundary line moves with the critical velocity a′ = λr and simultaneously either coincides with one of the r-characteristics of system (31) or is an envelope of the family of these r-characteristics arriving from the domain x > a(t), then the value of the r-invariant (47a) is brought to each point of the boundary line, and the velocity of motion of this line with allowance for condition (55) is determined by formula (57). It follows from here that such a flow is formed under the body lifting law ′

aH ′′ = (1 − Ha )λr rx+ − 2UH ′ − λr λs f ′ ,

Ha Px− = aH ′′ + (U 2 − Ha )f ′ + 2UH ′ = 0; for this reason, the equality Px = 0 should be satisfied for the boundary line to propagate along the r-characteristic of system (31). As is shown in Appendix B, if the boundary line propagates with the critical velocity a′ = λr and Px− ̸ = 0 on this line, then this

(

+

lim hx = lim

a′ →λr

a′ →λr

Ha Px−

′

f −

) = −∞,

(a′ − λr )(a′ − λs )

then the free surface of the liquid at a′ = λr and Px− ̸ = 0 approaches the lower surface of the body and makes a right angle with the horizontal line. 11. Energy conservation law in the domain adjacent to the body and motion of the boundary line under the condition Px− > 0 Let us assume that body lifting in the time interval (t1 , t2 ) ⊆ (T1 , T2 ) occurs under the condition Px− > 0, which admits the motion of the boundary line x = a(t) with the velocity a′ ≤ λr ; moreover, in the case of motion with the critical velocity a′ = λr , the boundary line is an envelope of the family of r-characteristics emanating from this line toward the domain x > a(t). In this case, the motion of the boundary line cannot be uniquely determined by formula (12) within the framework of the shallow water theory. Therefore it is necessary to use one more additional relation, which should be derived from the fundamental physical law of the liquid flow in the domain |x| ≤ a(t) adjacent to the body. Following Ostapenko and Kovyrkina [8], we use the energy conservation law in the domain adjacent to the body S(t) = {(x, z) : |x| ≤ a(t), 0 ≤ z ≤ h(x, t)}

(67)

as such a physical law. The total energy of the liquid in domain (67), which is the sum of the kinetic and potential energies, is determined by the formula a

∫

h

∫

E= −a

v 2 + w2

)

+ gz dzdx ) h( ) 2 2 2 v + w dz + gh dx.

= 0

(

2

0

∫ a (∫

(68)

0

The change in the total energy (68) results from its flux with the velocity

(66)

where rx+ = 0 in the case of boundary line propagation along the r-characteristic. As it follows from Appendix A, if the boundary line propagates with the supercritical velocity (65), then the condition of local correctness of such a flow (61) is equivalent to the validity of the inequality Px− > 0, which implies that the surface pressure ˜ p is negative in a certain left one-sided neighborhood (a(t) −ε, a(t)) of the boundary line. Thus, the body accelerates lifting of the liquid adjacent to it in this neighborhood. By virtue of formula (A.7), condition (66) at rx+ = 0 is equivalent to the condition

−

line is an envelope of the family of r-characteristics that move away from this line to the domain x > a(t) if the inequality Px− > 0 is satisfied and arrive on the boundary line from this domain if the opposite inequality Px− < 0 is satisfied. These inequalities are correlated with the condition of local correctness of solution (61), which implies that Px− > 0 if the boundary line moves with the supercritical velocity (65) and Px− < 0 if it moves with the subcritical velocity (42). As, by virtue of the well-posedness condition, formulas (A.3)–(A.7) yield

Ha

∫ V =2

ψ (a, z , t)dz ,

ψ = ( v − a′ )

(

v 2 + w2 2

0

) + gz ,

(69)

through the lateral boundaries x = ±a(t) of domain (67) and from the pressure work W1 = −2a′

Ha

∫

p(a, z , t)dz ,

0

W2 = −H ′

∫

a

−a

˜ pdx

(70)

on these lateral boundaries and on the top boundary z = h(x, t) of domain (67). We obtain the total energy conservation law from (68)–(70): dE dt

+ V = W1 + W2 .

Writing this equation in the dimensionless variables (26) and (27), we obtain the following form with accuracy to O(δ ): d˜ E dt

˜1 + W ˜2 . +˜ V + ∆E = W

(71)

O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

305

Here a

∫

edx,

˜ E=

˜ V = 2(U − a′ )

Ha

∫

0

(

U2 2

0

) + z dz = (U − a′ )ea , (72a, b)

e = h(u2 + h),

˜1 = −2a′ W

ea = e(a, t) = Ha U 2 + Ha ,

(

Ha

∫

)

(Ha − z)dz = −a′ Ha2 ,

(73)

(74)

0

˜2 = −2H ′ W

a

∫ 0

˜ pdx, ˜ p =˜ J =

∫ x˜ F (ξ , t)dξ , h(ξ , t) a

(75a, b)

and the function ˜ F is determined by formula (37). The variable ∆E involved in formula (71) is related to the change of the vortex flow energy of such part of the liquid, which does not satisfy the hydrostatic approximation (20) and is concentrated in small neighborhoods |x ± a(t)| < δ of the boundary lines x = ±a(t). The width of these neighborhoods 2δ can be neglected within the framework of the hydrostatic approximation of the shallow water theory. As the shallow water equations (31) represent a hyperbolic system of conservation laws with convex expansion [13,14], the energy transition ∆E is similar to the loss of the total energy on shock waves in the classical shallow water theory [9,15]. It is shown in Appendix C that Eq. (71), after conversions with taking into account formulas (72)–(75), can be written in the form (C.6), which will be used for obtaining an additional relation that allows one to determine the velocity of the boundary x = a(t) under the condition Px− > 0. Following Ostapenko and Kovyrkina [8], we use the assumption that the minimum possible part of the total energy of the liquid transforms to the vortex energy ∆E when the liquid leaves domain (67) as the basic criterion that allows one to determine the position of the boundary line under the condition Px− > 0. As a result we have from (C.6) under the constraint a′ ≤ λr that the liquid flow occurs under the condition

∆E = min Ha2 (U − a′ ) = Ha2 (U − λr ) = Ha2 C = Ha5/2 . a′ ≤λr

(76)

Hence, the boundary line x = a(t) propagates with the critical velocity a′ = λr , and the function a(t) is calculated by means of integrating the ordinary differential equation a′ + aH ′ /Ha +

√

Ha = 0,

Ha = f (a) + H .

(77a, b)

12. Results of numerical simulations 12.1. Formulation of the numerical initial–boundary-value problem In view of the symmetry of the problem with respect to the origin x = 0, the numerical solution of the homogeneous system of the shallow water equations (31) is sought in the domain [1, 4] at the first stage, in the domain [a(t), 4] with a moving left boundary at the second stage, and in the domain [0, 4] at the third stage. The right boundary of the computational domain at x = 4 is subjected to the non-reflecting boundary condition

Fig. 2. Wave profiles (a) and velocities (b) for four consecutive time instants in the case of wedge lifting with the velocity H ′ = 0.15 at the initial depths of the liquid under the wedge h1 = 0.25 and h2 = 0.75; t = 0.51 (1), 2.2 (2), 3.5 (3), and 5.0 (4).

The conditions imposed on the left boundaries of the computational domain are condition (33) at the first stage and the no-penetration condition u(0, t) = q(0, t) = 0 at the third stage. The conditions on the moving left boundary x = a(t) at the second stage depend on the sign of the one-sided pressure derivative Px− . If the inequality Px− ≤ 0 is satisfied, then one of the two conditions (59) is imposed on this boundary, and no boundary conditions are imposed if the opposite inequality Px− > 0 is satisfied. As the case f (x) = 0 corresponding to lifting a rectangular beam from shallow water was studied Ostapenko and Kovyrkina [8], in all calculations described further, the function f (x) defining the lower surface of the lifted body is chosen in two different ways f (x) = k1 |x|,

f (x) = k2 |x|3 ,

ki > 0.

(79a, b)

In the figures, the initial values of the depth are shown by the dashed curves, and the lower surface of the lifted body is depicted by the dot-and-dashed curves. For the numerical solution of system (31), we use a monotonic modification of the explicit second-order CABARET scheme [see 11]. The main advantages of this scheme are provided by the following property: in the linear case, this scheme is reversible in time and exact for two different Courant numbers r = 0.5 and r = 1. For this reason, the scheme possesses unique dissipative and dispersive properties. Moreover, the CABARET scheme has a compact stencil bounded by the size of one spatial–temporal cell of the difference grid. As a result, its accuracy does not decrease on severely nonuniform difference grids, which are used to construct the numerical algorithm at the second stage of the flow in the neighborhood of the moving boundary x = a(t) of the computational domain.

(78)

12.2. Flows where the liquid does not rise above the initial level h0 = 1

which means preservation of the r-invariant value brought from the unperturbed part x > 1 + t of the liquid. The boundary condition (78) allows approximate modeling of the solution of the initial–boundary-value problem in a semi-infinite spatial interval.

Figs. 2–7 illustrate the situations where the body is lifted from water with a constant velocity H ′ = 0.15 (Figs. 2 and 3) or with a constant acceleration H ′′ = 0.4 (Figs. 4 and 5) and H ′′ = 0.8 (Figs. 6 and 7). The flows formed in the case of wedge lifting (79a)

√

u(4, t) − 2 h(4, t) = r0 = −2,

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O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

Fig. 3. Wave profiles (a) and velocities (b) for four consecutive time instants in the case of parabolic body lifting with the velocity H ′ = 0.15 at the initial depths of the liquid under the body h1 = 0.25 and h2 = 0.75; t = 0.51 (1), 3.0 (2), 5.0 (3), and 5.7 (4).

Fig. 5. Wave profiles (a, b) and velocities (c, d) for six consecutive time instants in the case of parabolic body lifting with the acceleration H ′′ = 0.4 at the initial depths of the liquid under the body h1 = 0.25 and h2 = 0.75; t = 0.4 (1), 0.8 (2), 1.6 (3), 1.94 (4), 2.8 (5), and 3.5 (6).

Fig. 4. Wave profiles (a, b) and velocities (c, d) for six consecutive time instants in the case of wedge lifting with the acceleration H ′′ = 0.4 at the initial depths of the liquid under the wedge h1 = 0.25 and h2 = 0.75; t = 0.3 (1), 1.0 (2), 1.5 (3), 1.94 (4), 2.5 (5), and 3.5 (6).

are shown in Figs. 2, 4, and 6, and those for a body whose lower surface is defined by a cubic parabola (79b) are presented in Figs. 3, 5, and 7, where k1 = k2 = 0.5. The initial depths of water under the body are h1 = 0.25 and h2 = 0.75 in Figs. 2–5 and h1 = 0.5 and h2 = h0 = 1 in Figs. 6 and 7. Conditions (34) and (35) are satisfied for the solution shown in Figs. 2–5; the first condition ensures the existence of the first stage of the flow,

Fig. 6. Wave profiles for six consecutive time instants in the case of wedge lifting with the acceleration H ′′ = 0.8 at the initial depths of the liquid under the wedge h1 = 0.5 and h2 = 1; t = 0.3 (1), 0.6 (2), 0.9 (3), 1.12 (4), 2.0 (5), and 3.0 (6).

and the second condition guarantees the well-posedness of the initial–boundary-value problem (31)–(33) at this stage. As the end faces of the body for the solution illustrated in Figs. 6 and 7 are not immersed in water, the first stage of the process of body lifting is absent, and the flow begins from the second stage. In the case of body lifting with the constant velocity H ′ = 0.15, the velocity of the liquid under the body becomes finite immediately after the beginning of lifting, at the time t = 0 + 0 (dotted curves in Figs. 2b and 3b). The first stage is finalized when the end faces of the body completely leave the water medium (curves 1 in

O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

307

Table 1 Maximum levels of the liquid obtained in the case of lifting of a wedge and a parabolic body (79), where k1 = k2 = 1/2, with different values of the acceleration; initial depths under the body h1 = 0.5 and h2 = 1. H ′′ Wedge Parabolic body

Fig. 7. Wave profiles for six consecutive time instants in the case of parabolic body lifting with the acceleration H ′′ = 0.8 at the initial depths of the liquid under the body h1 = 0.5 and h2 = 1; t = 0.3 (1), 0.7 (2), 1 (3), 1.12 (4), 2 (5), and 3 (6).

Fig. 8. Trajectory of motion of the boundary line a(t) in the case of wedge lifting (curves 1–3) and parabolic body lifting (curves 4–6) with the velocity H ′ = 0.15 at the initial depths of the liquid under the body h1 = 0.25 and h2 = 0.75 (1, 4), with the acceleration H ′′ = 0.4 at the initial depth h1 = 0.25 and h2 = 0.75 (2, 5), and with the acceleration H ′′ = 0.8 at the initial depth h1 = 0.5 and h2 = 1 (3, 6).

Figs. 2a,b and 3a,b), which occurs at the time T1 ≈ 0.51, and the second stage (curves 2 and 3 in Fig. 2a,b and curves 2 in Fig. 3a,b) is finalized at the time T2 = 5, when the body is separated from water (curves 4 in Fig. 2a,b and curves 3 in Fig. 3a,b). After body separation from water, two rather flat diverging elevation waves are formed (curves 4 in Fig. 3a,b). In the case of body lifting with the constant acceleration H ′′ = 0.4, the first stage (curves 1 in Figs. 4a,c and 5a,c) is finalized at the time T1 ≈ 0.44, and the second stage (curves 2 and 3 in Figs. 4a,c and 5a,c) is finalized at the time T2 ≈ 1.94 (curves 4 in Figs. 4b,d and 5b,d). In the case of body lifting with the constant acceleration H ′′ = 0.8, where the first stage is absent, the second stage (curves 1–3 in Figs. 6a and 7a) is finalized at the time T2 ≈ 1.12. In the case of body lifting with the constant acceleration H ′′ = 0.4 or H ′′ = 0.8, wave formation at the second stage is much more intense than that in the case of body lifting with the constant velocity H ′ = 0.15. Therefore, the elevation waves shown by curves 4–6 in Figs. 4b–7b have much steeper profiles at the third stage, leading to the formation of shock waves (curves 6 in Figs. 5b,d and 7b).

1.5 1.09 1.07

3.0 1.37 1.34

4.5 1.58 1.54

6.0 1.76 1.72

7.5 1.91 1.86

Fig. 8 shows the function x = a(t) defining the equation of motion of the boundary line for the solution depicted in Figs. 2– 7. In Fig. 8 curves 1, 4 illustrate body lifting with the velocity H ′ = 0.15, curves 2, 5 illustrate body lifting with the acceleration H ′′ = 0.4, and curves 3, 6 illustrate body lifting with the acceleration H ′′ = 0.8. The bold and thin curves correspond to lifting of a wedge and a parabolic body from water, respectively. At the second stage of the solution shown in Figs. 2–5 and 7, the pressure derivative is Px− < 0, the boundary line propagates with the subcritical velocity (42), and the r-characteristics arriving on this line bring the initial value r0 = −2 of the r-invariant from the unperturbed part of the external flow. As a result, the level of the liquid in the domain |x| < a(t) where it is adjacent to the body does not rise above the initial level of the liquid h0 = 1 outside the body, and body separation from the liquid occurs at the depth h(0, T2 ) = H(T2 ) = h0 = 1. In the case of the solution shown in Fig. 6, the pressure derivative in a certain time interval (T1 , t2 ) ⊂ (T1 , T2 ) at the beginning of the second stage is Px− > 0; hence, the boundary line propagates with the critical velocity a′ = λr (dashed part of curve 3 in Fig. 8) and is an envelope of the family of r-characteristics emanating from this line to the domain x > a(t). In the time interval (t2 , T2 ) the derivative is Px− < 0, the boundary line propagates with the subcritical velocity (bold solid part of curve 3 in Fig. 8), and the r-characteristics that left this line at t ∈ (T1 , t2 ) return to this line within a certain time interval (t2 , t3 ) ⊂ (t2 , T2 ) before the second stage is finalized. As a result, in the time interval (t3 , T2 ), the r-characteristics arriving on the boundary line bring the initial value r0 = −2 of the r-invariant; therefore, body separation from the liquid also occurs at the depth h(0, T2 ) = h0 = 1. 12.3. Flows where the liquid rises above the initial level h0 = 1 at the end of the second stage Figs. 9–12 illustrate the situations where the body is lifted from water with the constant acceleration H ′′ = 1.5 (Figs. 9, 10) and H ′′ = 6 (Figs. 11, 12) at the initial depth of water under the body being h1 = 0.5 and h2 = 1. As h2 = h0 , the first stage of the process is absent. The flows formed in the case of wedge lifting (79a) are shown in Figs. 9 and 11; those in the case of the parabolic body (79b) are presented in Figs. 10 and 12, where k1 = k2 = 0.5. In this variant of body lifting (similar to the case illustrated in Fig. 6), the pressure derivative is Px− > 0 in a certain time interval (T1 , t2 ) ⊂ (T1 , T2 ) and Px− < 0 in the time interval (t2 , T2 ). In the time interval (T1 , t2 ), the boundary line x = a(t) propagates with the critical velocity a′ = λr (dashed parts of the curves in Fig. 13) and is an envelope of the family of r-characteristics moving away from this line to the external domain x > a(t). These r-characteristics form the domain Sa = (a(t), a(t) + c(t)) of influence of the boundary line, where the flow is completely determined by the flow parameters on the boundary line and is independent of the external flow at x > c(t). Inside the domain Sa , the absolute value of the liquid velocity decreases (curves 1–3 in Figs. 9c, 10c) with simultaneous reduction of its depth (curves 1– 3 in Figs. 9a, 10a), which is typical for supercritical flows in the coordinate system moving with the velocity of the boundary line.

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Fig. 11. Wave profiles for six consecutive time instants in the case of wedge lifting with the acceleration H ′′ = 6 at the initial depths h1 = 0.5 and h2 = 1; t = 0.1 (1), 0.3 (2), 0.6 (3), 0.65 (4), 0.9 (5), and 1.2 (6).

Fig. 9. Wave profiles (a, b) and velocities (c, d) for six consecutive time instants in the case of wedge lifting with the acceleration H ′′ = 1.5 at the initial depths of the liquid under the wedge h1 = 0.5 and h2 = 1; t = 0.2 (1), 0.5 (2), 0.7 (3), 0.89 (4), 1.2 (5), and 1.6 (6). Fig. 12. Wave profiles for six consecutive time instants in the case of parabolic body lifting with the acceleration H ′′ = 6 at the initial depths h1 = 0.5 and h2 = 1; t = 0.1 (1), 0.3 (2), 0.6 (3), 0.64 (4), 0.9 (5), and 1.2 (6).

Fig. 10. Wave profiles (a, b) and velocities (c, d) for six consecutive time instants in the case of parabolic body lifting with the acceleration H ′′ = 1.5 at the initial depths of the liquid under the body h1 = 0.5 and h2 = 1; t = 0.2 (1), 0.5 (2), 0.7 (3), 0.87 (4), 1.2 (5), and 1.6 (6).

In the time interval (t2 , T2 ), the boundary line propagates with the subcritical velocity (solid parts of the curves in Fig. 13), and the r-characteristics from the external domain x > a(t) arrive on this line. However, in contrast to the case illustrated in Fig. 6, all these r-characteristics leave the boundary line at t ∈ (T1 , t2 ); therefore, they do not bring any information from the unperturbed region of the solution located in the domain x > L + c0 t = 1 + t. In view of this fact, the body separates from the liquid (curves 4 in Figs. 9, 10, 11, 12b) at the depth h(0, T2 ) > h0 = 1, i.e., the depth of the liquid rising behind the body at the second stage becomes greater than the initial depth of the liquid outside the body. Moreover, the maximum height of liquid lifting increases significantly with increasing acceleration of body lifting (see Table 1). As is seen from this table, in the case of lifting the wedge (79a) and the parabolic body (79b) with an identical acceleration, the maximum height of liquid lifting is slightly greater in the case of the wedge. Because of liquid lifting near the origin above the initial level h0 , two diverging waves are formed at the third stage, and the liquid depth in these waves is greater than h0 . In the wave propagating in the positive direction of the x axis (curves 5 and 6 in Figs. 9b, 10b), the liquid velocity is positive (curves 5 and 6 in Figs. 9d, 10d); outside this wave, at x > 0, the liquid velocity is negative or equal to zero. It follows from Figs. 9–12 that the amplitude of these waves increases with increasing acceleration of body lifting, and the shock is formed in the frontal part of these waves owing to a gradient catastrophe (curves 6 in Figs. 11b and 12b).

O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

309

Fig. 13. Trajectory of motion of the boundary line a(t) in the case of parabolic body lifting with the accelerations H ′′ = 1.5 (1), 3 (2), and 6 (3) at the initial depths h1 = 0.5 and h2 = 1.

Fig. 14. The volume of liquid V (t) rising above the initial level h0 in the case of lifting from water the parabolic body (solid lines 1, 3) and the wedge (dashed lines 2, 4) with the accelerations H ′′ = 1.5 (lines 1, 2) and H ′′ = 6 (lines 3, 4). The arrows show the time moments at which bodies separate from the liquid.

Fig. 15. Wave profiles for three consecutive time instants obtained in the case of lifting of parabolic bodies (13.3b) from water with the acceleration H ′′ = 3 at the initial depth h2 = 1: k2 = 0.4, and T2 = 0.74 (a); k2 = 0.2 and T2 = 0.72 (b); k2 = 0.01 and T2 = 0.71 (c); t = 0.5 (1), T2 (2), and 2 (3).

Fig. 14 shows the graph of function

∫ V (t) = 2

+∞

h+ (x, t)dx,

0

h+ =

{ ∆ h, 0,

∆h > 0, ∆h ≤ 0,

(80)

∆ h = h − h0 ,

that defines the volume of liquid rising above the initial level h0 in the case of lifting from water the wedge and the parabolic body presented in the Figs. 9–12. It follows from Fig. 14 that the volume V (t) increases significantly with increasing acceleration of the body lifting and that this volume is somewhat larger when lifting the wedge than when lifting the parabolic body. It can be seen from this figure that at some initial time interval (t3 , t4 ) ⊂ (T1 , T2 ) the value of V (t) sharply increases from zero to some local maximum, due to the rise of the wetted part of the lower body surface above the initial liquid level h0 (lines 3 in Figs. 11a and 12a). Wherein the length of the time interval (t3 , t4 ) is significantly less in case of lifting the parabolic body, since its lower surface quickly flattens when x → 0. In the next time interval (t4 , t5 ) ⊂ (T1 , T2 ), which right boundary is close to the time moment T2 , the function V (t) decreases monotonically, since on this interval the pressure derivative is Px− (t) < 0 (see Fig. 13) and the body decelerates lifting of the adjacent liquid in some neighborhood of the boundary line x = a(t). At t > t5 the function V (t) begins to increase again. This increase continues after the separation of the body from the liquid at t > T2 , which corresponds to the initial stage of the formation of the level elevation wave and its propagation in the positive direction of the x-axis (lines 5 in Figs. 9a–12a). Over time, the amplitude of this elevation wave gradually decreases and the function V (t) begins to decrease. As a result, the maximum value of the liquid volume rising above the initial level h0 achieved at some time moment T3 > T2 after separation of the body from the liquid and the value of T3 increases when the acceleration of body lifting decreases.

Fig. 16. The volume of liquid V (t) rising above the initial level h0 in the case of lifting from water with the acceleration H ′′ = 3 at the initial depth h2 = 1 the parabolic body with the coefficients k2 = 0.4 (1), k2 = 0.2 (2), and k2 = 0.01 (3). The arrows show the time moments at which bodies separate from the liquid.

Fig. 15 shows the liquid depth profiles obtained for lifting of parabolic bodies (79b) with the coefficients k2 = 0.4 (Fig. 15a), k2 = 0.2 (Fig. 14b), and k2 = 0.01 (Fig. 15c) from water with the constant acceleration H ′′ = 3. It follows from this figure that the maximum height of liquid lifting increases as the body becomes flatter: 1.34, 1.51, 1.67,

{ h(0, T2 ) = H(T2 ) =

k2 = 0.4, k2 = 0.2, k2 = 0.01.

Fig. 16 shows the volume V (t) of liquid rising above the initial level corresponding to the flows in Fig. 15. From Fig. 16 it follows that with the decrease in the coefficient k2 the values of the function V (t) monotonically increase for all t for which V (t) > 0. This means that the maximum volume of liquid above its initial level will be achieved when lifting a convex body with a

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Fig. 17. Pressure ˜ p(x, t) in the body–liquid contact region for four consecutive time instants in the case of wedge (a) and parabolic body (b) lifting with the velocity H ′ = 0.15 at the initial depths h1 = 0.25 and h2 = 0.75; t = T1 = 0.51 (1), 1.5 (2), 2.5 (3), and 3.5 (4).

Fig. 19. Pressure ˜ p(x, t) in the body–liquid contact region for five consecutive time instants in the case of wedge (a) and parabolic body (b) lifting with the acceleration H ′′ = 0.8 at the initial depths h1 = 0.5 and h2 = 1; t = 0.1 (1), 0.3 (2), 0.5 (3), 0.7 (4), and 0.8 (5).

Fig. 18. Pressure ˜ p(x, t) in the body–liquid contact region for three consecutive time instants in the case of wedge (a) and parabolic body (b) lifting with the acceleration H ′′ = 0.4 at the initial depths h1 = 0.25 and h2 = 0.75; t = T1 = 0.44 (1), tm (2), and 1.5 (3); tm = 0.80 (a) and tm = 0.89 (b), where tm is the time instant when the pressure reaches the maximum value at the second stage.

Fig. 20. Pressure ˜ p(x, t) in the body–liquid contact region for four consecutive time instants in the case of wedge (a) and parabolic body (b) lifting with the acceleration H ′′ = 1.5 at the initial depths h1 = 0.5 and h2 = 1; (a): t = 0.2 (1), t = 0.3 (2), tm = 0.5 (3), and t = 0.65 (4); (b): t = 0.2 (1), t = 0.3 (2), tm = 0.55 (3), and t = 0.7 (4), where tm is the time instant when the pressure reaches the maximum value at the second stage.

gently sloping lower surface. However, this surface cannot be flat, because during mathematical modeling of this problem within the framework of an ideal liquid flow, when a rectangular beam is lifted from its surface, no fluid will rise behind it.

lifting is illustrated in Figs. 17a–20a, and parabolic body lifting is shown in Figs. 17b–20b. The dashed curves in Figs. 17 and 18 show the pressure at the initial time t = 0. The dotted curves in Figs. 17–20 show the pressure at the time t = 0 + 0, immediately after the beginning of body lifting. It follows from formulas (36)–(38), (40) that the surface pressure ˜ p at the second stage at |x| ≤ a(t) can be represented as

12.4. Changes in pressure in the body–liquid contact region Figs. 17–20 show the pressure ˜ p(x, t) in the body–liquid contact region as a function of the spatial coordinate x ∈ [0, 1] for several consecutive times. The body is lifted with the constant velocity H ′ = 0.15 (Fig. 17), with the constant acceleration H ′′ = 0.4 (Fig. 18), with the constant acceleration H ′′ = 0.8 (Fig. 19), and with the constant acceleration H ′′ = 1.5 (Fig. 20). Wedge

˜ p(x, t) =

∫

x a(t)

˜ F (ξ , t)dξ h(ξ , t)

= p(x, t) + J(x, t),

where p(x, t) = f (a(t)) − f (x) is the hydrostatic component of the pressure corresponding to the case of a quiescent body and liquid,

O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

Fig. 21. Changes in the maximum pressure ˜ pm (t) (81) in the interval (0, a(t)) in the case of wedge (1, 3) and parabolic body (2, 4) lifting from water with the constant accelerations H ′′ = 0.4 (1, 2) and H ′′ = 0.8 (3, 4).

and

∫

J(x, t) =

x a(t)

F (ξ , t)dξ h(ξ , t)

,

F = xH ′′ + u2 f ′ + 2uH ′ ,

is its dynamic component arising due to body and liquid motion. Therefore, under the condition |J | ≪ 1, the surface pressure ˜ p coincides with its hydrostatic component p with accuracy to O(J). In the case of sufficiently slow lifting of the body, where H ′′ ≪ 1,

H′ ≪ 1

|u| ≪ 1 ∀ |x| < a,

⇒

the inequality |J | ≪ 1 is satisfied at the entire second stage of the process. Such a situation is illustrated in Fig. 17, where the body is lifted from water with the constant velocity H ′ = 0.15. As a result, at the second stage we have ˜ p ≈ pw = (a − x)/2 for wedge lifting (curves 1–4 in Fig. 17a) and ˜ p ≈ pb = (a3 − x3 )/2 for parabolic body lifting (curves 1–4 in Fig. 17b). If the body is lifted with the finite velocity or acceleration, i.e., H ′ = O(1) or H ′′ = O(1), then the hydrodynamic and dynamic components of the pressure ˜ p are comparable at the beginning of the second stage, i.e., J = O(˜ p). In this case, however, we obtain the following estimates when the second stage is finalized, at 0 < x < a(t) ≪ 1, with allowance for (11): pw = O(a − x),

pb = O a2 (a − x) ,

(

)

F = O(x) ⇒ J = O (a(a − x)) This means that the surface pressure ˜ p at |a(t)| ≪ 1 is close to its hydrodynamic component ˜ p ≈ pw in the case of wedge lifting (curve 3 in Fig. 18a, curves 4 and 5 in Fig. 19a, and curves 3 and 4 in Fig. 20a) and close to its dynamic component ˜ p ≈ J in the case of parabolic body lifting (curve 3 in Fig. 18b, curves 4 and 5 in Fig. 19b, and curves 3 and 4 in Fig. 20b). It follows from formulas (38)–(41) that the surface pressure ˜ p(x, t) in the case of lifting of a rectangular beam (f (x) = 0) is a monotonic function of the variable x ∈ (0, a(t)): rigorously decreasing at G > 0 and rigorously increasing at G < 0. However, in the case of lifting a body with f (x) ̸ = 0, the surface pressure behaves in a more complicated manner. In the case the wedge and the parabolic body lifting with the constant velocity H ′ = 0.15 (Fig. 17) or wedge lifting with the constant acceleration H ′′ = 0.4 (Fig. 18a), the pressure ˜ p(x, t) rigorously monotonically decreases at x ∈ (0, a(t)) and t ∈ (T1 , T2 ). However, in all other cases shown in Figs. 18b, 19, and 20, the pressure behavior is not monotonic. In particular, the pressure function ˜ p(x, t) may have local and absolute extreme points along the x axis inside the interval (0, a(t)). Fig. 21 shows the evolution of the maximum pressure in the interval (0, a(t))

˜ pm (t) = max ˜ p(x, t) x∈(0,a(t))

(81)

311

Fig. 22. Time evolution of the coordinate of the point xm (t) where the maximum pressure is reached in the case of parabolic body lifting with the acceleration H ′′ = 0.4 at the initial depths h1 = 0.25 and h2 = 0.75 (1) and with the acceleration H ′′ = 0.8 at the initial depths h1 = 0.5 and h2 = 1 (2).

in the case of wedge and parabolic body lifting with the acceleration H ′′ = 0.4 at the initial depth h1 = 0.25 and h2 = 0.75 and with the acceleration H ′′ = 0.8 at the initial depth h1 = 0.5 and h2 = 1. It follows from these graphs that the behavior of the maximum pressure (81) is essentially nonmonotonic as a function of time. In the case of wedge lifting, the maximum pressure is reached at the point xm = 0, where the wedge is immersed in water to the maximum extent. In the case of parabolic body lifting, the coordinate of the point xm (t) depends on time (Fig. 22), and the maximum pressure can be reached inside the interval (0, a(t)). 13. Conclusion One of the main difficulties arising in modeling the considered problem within the framework of the system of equations of the first approximation of the shallow water theory is related to justification of the applicability of these equations in domain (6) of the liquid contact with the lower surface of the lifted body. As the upper boundary of the liquid is adjacent to the lower surface of the body in domain (6) and its motion is completely determined by the body lifting law (7), it is impossible to introduce the characteristics length L0 of free surface waves in this domain; hence, the classical long-wave approximation (21) cannot be used. This is most clearly manifested in the case of lifting of a beam [see 8] for which f (x) = 0, and the upper boundary of the liquid in domain (6) is horizontal. The system of shallow water equation (30) derived within the framework of the long-wave approximation (21) is usually applied [see 9] for determining the depth and horizontal velocity of the liquid in modeling wave flows in open riverbeds with allowance for the wind action, which is taken into account by means of setting the surface pressure function ˜ p. In our problem, however, system (30) is used in an essentially different way: for a given law of body lifting, Eq. (30a) is used to determine the flow rate and velocity of the liquid in domain (6), and Eq. (30b) is used to find the pressure ˜ p on the body–liquid boundary. Moreover, in the standard application of system (30) for modeling wave flows in open channels, the velocity of propagation of small perturbations is finite and is determined by the velocity characteristics λr and λs , but the use of this system for modeling liquid flows in domain (6) under the lifted body leads to a infinite velocity of propagation of small perturbations. As a result, the liquid velocity (11) and surface pressure (38) instantaneously change in the entire domain (6) at the beginning of body lifting. To justify the applicability of the first approximation of the shallow water theory for the considered problem, the basic conservation laws of this theory are derived from two-dimensional

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integral conservation laws of mass and momentum, which describe a plane-parallel flow of an ideal incompressible liquid above a horizontal bottom. As the integral mass conservation law (4) for the shallow water model is derived without constraints on the two-dimensional flow parameters, the resultant formulas (9) and (11) for the flow rate and vertically averaged liquid velocity are exact and are valid during the first two stages, including the initial time interval of relaxation when the local hydrostatic approximation condition (20) is not yet satisfied. If conditions (18) are satisfied, the total momentum conservation law (17) for the shallow water model follows from the two-dimensional horizontal momentum conservation law (13) with accuracy to O(J1 + J2 ). To justify the applicability of the conservation law (17) for the approximate solution of our problem, we propose to derive it with the use of a special hydrostatic approximation (20), which is a generalization of the long-wave approximation; in contrast to the latter, this hydrostatic approximation is local and remains meaningful in domain (6) under the body. The use of the local hydrostatic approximation for writing the considered problem in the dimensionless variables (25)–(27) leads to natural constraints (28) on the dimensional velocity and acceleration of body lifting. As the flow rate on the body boundary at the first stage is independent of the function f (x) defining the lower surface of the body, the solution of the initial–boundary-value problem (31)– (33), which describes the liquid flow in the external domain x > a(t) at the first stage, is also independent of the form of the function f (x). Thus, this problem was analyzed in much detail in Ostapenko and Kovyrkina [8], where lifting of a rectangular beam from shallow water was considered, and conditions (34) and (35) of well-posedness of this problem were obtained. As we had to use the heuristic assumption [4] that the velocity of the liquid–body contact boundary is proportional to the local velocity of the liquid flow in theoretical modeling of lifting of a flat body from the free boundary of an infinitely deep liquid, the main challenge of this work was to obtain an exact law of motion of this boundary line within the framework of the hydrostatic approximation in solving the problem of lifting of a symmetrical convex body partly immersed into shallow water. As a result of solving this problem, it was shown that the motion of the boundary line x = a(t) essentially depends on the sign of the left one-sided spatial derivative of the pressure Px− on this line. The sign of this derivative determines the sign of the surface pressure ˜ p in a certain left one-sided neighborhood (a(t) −ε, a(t)) of the boundary line. As a result, the body decelerates lifting of the liquid adjacent to the body in this neighborhood at Px− < 0 and accelerates the lifting process at Px− > 0. Under the condition Px− < 0, the position of the boundary line x = a(t) is determined from the algebraic equation (48a), and the velocity of its propagation is calculated by formula (57) if the additional inequality (55) is satisfied. Under the condition Px− ≥ 0, the boundary line propagates with the critical velocity a′ = λr and its position is determined by integrating Eq. (77a). The inequality Px− < 0 should be satisfied for the boundary line to propagate with the subcritical velocity (42), and the equality Px− = 0 should be valid for the boundary line to propagate along the r-characteristic of system (31). If the inequality Px− > 0 is satisfied, then the boundary line propagates with the critical velocity a′ = λr and is an envelope of the family of r-characteristics emanating from this line to the external domain x > a(t). If the inequality Px− < 0 is satisfied and the boundary line propagates with the critical velocity, then it is an envelope of the family of r-characteristics arriving on this line from the domain x > a(t).

As the motion of the boundary line under the condition Px− > 0 cannot be uniquely determined within the framework of the shallow water theory, it is necessary to use one more additional relation, which should be derived from the fundamental physical law of the considered liquid flow. Following Ostapenko and Kovyrkina [8], we use the energy conservation law (71) in the domain |x| < a(t) adjacent to the body as such a physical law. As the basic criterion that allows one to determine the position of the boundary line in this case, we use the assumption that the minimum possible part of the total energy of the liquid transforms to the vortex energy ∆E when the liquid leaves domain (67). As a result we have from (76) that the boundary line propagates with the critical velocity a′ = λr . If the inequality Px− ≥ 0 is satisfied at the second stage, which means that the boundary line propagates with the subcritical or critical velocity a′ ∈ [λr , λs ), and the r-characteristics arriving on this line bring the initial value r0 = −2 of the r-invariant from the unperturbed part of the external flow, then the liquid layer in the domain |x| < a(t) where it is adjacent to the body does not rise above the initial level of the liquid h0 = 1 outside the body; as a result, body separation from the liquid occurs at the depth h(0, T2 ) = h0 = 1 (Figs. 2–7). If the inequality Px− < 0 is satisfied at the beginning of the second stage, the maximum height of liquid lifting behind the body can be appreciably greater than the initial liquid depth outside the body (Figs. 9–12, 15). It is further planned to compare the solutions obtained in this work with numerical simulations performed on the basis of hydrodynamic models of higher levels. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Correctness of the solution in the case of boundary line motion with the subcritical or supercritical velocity If the boundary line x = a(t) propagates with the subcritical velocity (42) or supercritical velocity (44), solution (59) known on this line can be unambiguously continued into the external domain x > a(t). In particular, it is possible to determine the one-sided temporal and spatial derivatives (52) of the functions h(x, t) and u(x, t) on this line. Differentiating relations (59) with respect to t at x = a(t) + 0, we obtain ′ + ′ h+ t + a hx = Ha ,

′ + ′ u+ t + a ux = U .

(A.1)

+ Eliminating the temporal derivatives h+ t and ut from system (A.1) by using Eq. (31a) and the differential corollary

ut + uux + hx = 0 of system (31), we obtain the following algebraic equations for determining the spatial derivatives: + ′ (a′ − U)h+ x − Ha ux = Ha ,

′ + ′ − h+ x + (a − U)ux = U .

(A.2)

Solving system (A.2) by Cramer’s rule and taking into account (12), (42), and (45), we obtain the expressions for the spatial derivatives h+ x =

∆1 , ∆

u+ x =

∆2 , ∆

(A.3)

O.A. Kovyrkina and V.V. Ostapenko / European Journal of Mechanics / B Fluids 79 (2020) 297–314

where the determinants are calculated by the formulas

∆ = (a − U) − Ha = (a − U) − C = (a − λr )(a − λs ), (A.4) 2

′

2

′

2

′

′

∆1 = (a′ − U)Ha′ + Ha U ′ = (a′ − 2U)Ha′ − (aH ′ )′

= f ′ (a′ )2 − 2U(H ′ + f ′ a′ ) − aH ′′ ,

(A.5)

∆2 = (a′ − U)U ′ + Ha′ .

As it follows from (B.2) and (B.3), if the boundary line propagates with the critical velocity a′ = λr and the condition Px− ̸ = 0 is satisfied on this line, then this line is an envelope of the family of r-characteristics, which emanate from the boundary line to the domain x > a(t) under the condition Px− > 0 and arrive on the boundary line from this domain at Px− < 0. Appendix C. Derivation of the formula for ∆E

In view of formulas (A.3)–(A.5) and of the fact that λr + λs = 2U, the local correctness condition (61) can be written as ′ f ′ − h+ x = f −

f ′ (a′ )2 − 2U(H ′ + f ′ a′ ) − aH ′′ (a′

)(a′

(A.6)

On the other hand, in view of formulas (36), (37), and (46) and of the fact that U − Ha = U − C = (U − C )(U + C ) = λr λs , 2

2

2UH ′ + aH ′′ + λr λs f ′ Ha

.

(A.7)

It follows from here that the correctness condition (A.6) can be represented as f ′ − h+ x =

Ha Px− (a′

− λr )(a′ − λs )

(C.1)

obtained by means of subtracting Eq. (30a) multiplied by u from Eq. (30b) and subsequent division by h. Summing up Eq. (30b) multiplied by u and Eq. (C.1) multiplied by q, we obtain the equation (qu)t + (qu2 )x + 2qhx = −2q˜ px ,

(C.2)

which describes the change in the doubled kinetic energy of the flow qu = hu2 . Adding Eq. (30a) multiplied by 2h to Eq. (C.2), we obtain the equation

we obtain Px− (t) =

The differential corollary of system (30) is the equation ut + (u2 /2 + h)x = −˜ px ,

=

− λr − λs ) 2UH ′ + aH ′′ + λr λs f ′ > 0. (a′ − λr )(a′ − λs )

2

313

> 0,

which is equivalent to the inequality Px− (t) < 0 in the case of boundary line motion with the subcritical velocity (42) and to the inequality Px− (t) > 0 in the case of boundary line motion with the supercritical velocity (44).

et + u(e + h2 )

(

) x

= −2q˜ px ,

(C.3)

which describes the change in the doubled specific energy of the flow e = h(u2 + h). In view of Eq. (C.3) and formula (9), the expression for the temporal derivative of the function ˜ E defined by formula (72a) is d˜ E dt

=

d dt

a

∫

edx = ea a′ +

a

∫

et dx 0

0

= e a a′ −

a

∫

u(e + h2 )

] + 2q˜ px dx = 0 ∫ a ( ) ′ x˜ px dx, ea a − U ea + Ha2 + 2H ′

Appendix B. Conditions at which the boundary line is an envelope of the family of r-characteristics

[(

)

x

(C.4)

0

If the boundary line x = a(t) propagates with the critical velocity a′ = λr , then it moves with the acceleration

Applying the formula of integration by parts and taking into account the condition ˜ p(a, t) = 0, we obtain

a′′ = λ′r = U ′ − C ′

∫

(B.1)

where U = −aR, R = H ′ /Ha , C = formulas

√

Ha , Ha = H + f (a). Using the

(

C′ =

H ′′ Ha − H ′ Ha′

=

Ha2 H′

√a

=

2 Ha H

′

√

Ha

H ′′ Ha − H ′ (H ′ + f ′ a′ ) Ha2

H ′ + f ′ a′ 2C H

H′

⇒

′

C

)

=

= 2C ′ −

H ′′ − Rf ′ a′ Ha

f ′ a′ C

− R2 ,

f a

= = 2C − , Ha C C we can represent the derivative (B.1) in the form

CR =

a′′ = C ′ −

2UH ′ + aH ′′ + λr λs f ′ Ha

= C ′ − Px− .

(B.2)

If the point (a(t), t) of the boundary line x = a(t) lies on the r-characteristic of system (31), where the r-invariant has a constant value r1 , the liquid and sound velocities U = U(a(t), t) and C = C (a(t), t) at this point satisfy the condition U = 2C + r1 , whereas the r-characteristic velocity at this point is calculated by the formula λr = U − C = C + r1 . From here, the r-characteristic acceleration on the boundary line is

λ′r = (C + r1 )′ = C ′ .

0

xd˜ p = (x˜ p)|a0 −

a

∫ 0

a

∫ ˜ pdx = −

0

˜ pdx.

(C.5)

∆E = Ha2 (U − a′ ).

(C.6)

References

,

′ ′

′

0

a

∫ x˜ px dx =

Substituting (C.5) into (C.4), and then substituting (C.4), (72b), (74), and (75a) into Eq. (71) and applying reduction, we find that ∆E is calculated by the formula

U ′ = −aR′ − Ra′ = (C − U)R − Ra′ = a R2 − R′ + CR, R′ =

a

(B.3)

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