Extrema of the dynamic pressure in an irrotational regular wave train A Constantin Citation: Phys Fluids 28, 113604 (2016); doi: 10.1063/1.4967362 View online: http://dx.doi.org/10.1063/1.4967362 View Table of Contents: http://aip.scitation.org/toc/phf/28/11 Published by the American Institute of Physics Articles you may be interested in Effects of wing shape, aspect ratio and deviation angle on aerodynamic performance of flapping wings in hover Phys Fluids 28, 111901111901 (2016); 10.1063/1.4964928 An accelerated stochastic vortex structure method for particle collision and agglomeration in homogeneous turbulence Phys Fluids 28, 113301113301 (2016); 10.1063/1.4966684 Vortex ring formation in starting forced plumes with negative and positive buoyancy Phys Fluids 28, 113601113601 (2016); 10.1063/1.4966648 Vortex identification from local properties of the vorticity field Phys Fluids 29, 015101015101 (2017); 10.1063/1.4973243 PHYSICS OF FLUIDS 28, 113604 (2016) Extrema of the dynamic pressure in an irrotational regular wave train A Constantina) Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria (Received 28 August 2016; accepted 26 October 2016; published online 14 November 2016) We prove that in a regular wave train, which propagates at the surface of water in irrotational flow over a flat bed, the maximum and minimum of the dynamic pressure occur at the wave crest and at the wave trough, respectively This result is valid without any restrictions on the wave amplitude Published by AIP Publishing [http://dx.doi.org/10.1063/1.4967362] I INTRODUCTION Pressure gradients can drive fluid motion but this principle fails for a static fluid, in which case the fluid pressure is hydrostatic, counteracting the force of gravity so that the fluid is in a state of equilibrium This consideration motivates the decomposition of the pressure beneath a surface water wave as the sum of a dynamic part that encodes the fluid motion and a hydrostatic part due to the weight of the fluid A detailed understanding of the behaviour of the dynamic pressure in a water-wave flow is not merely of academic interest since a better pressure field knowledge in wave flows is very important in estimating the forces acting on maritime structures and most wave measurements in the open sea are carried out by means of submerged pressure transducers — the notable advantages being the affordable gathering of precise data due to sturdiness and small manufacturing and maintenance costs.3,7,12,17,22 Of particular interest are waves of large amplitude, for which nonlinearity is essential While for waves of small amplitude linear theory is valid, most waves relevant in ocean engineering not fall into this category This prompts the need for nonlinear wave theories Perturbation methods give insight into the dynamics of weakly nonlinear flow regimes20,23 but to address the physically most interesting setting of waves of large amplitude it is advisable to pursue a study of the governing equations In order to make progress, it is necessary to take advantage of special hypotheses which suggest themselves on the basis of the contemplated physical circumstances More precisely, for surface water waves of large amplitude the effects of viscosity and surface tension are negligible, so that in the absence of stratification the governing equations are the incompressible Euler equations for a homogeneous fluid with a free-boundary, acted upon by gravity Moreover, since most ocean motions have slowly varying wave characteristics, over moderate length scales the wave field can be described accurately by a uniform nonlinear wave train propagating in a fixed direction (for example, this is typical for swell in the open ocean, far from the area where the sea waves were generated by a storm), with the absence of strong underlying currents captured by the assumption of irrotational flow Furthermore, since abyssal plains cover about 40% of the ocean floor, the assumption of a flat bed is of practical relevance In this paper, we will investigate the dynamic pressure beneath a regular periodic wave train propagating in irrotational flow over a flat bed Physical intuition, field data, and numerical simulations offer insight into the fluctuations of the dynamic pressure For example, laboratory measurements24,25 and numerical simulations2,21 indicate that the maximal values of the dynamic pressure are encountered in the region below the a) Electronic mail: adrian.constantin@univie.ac.at 1070-6631/2016/28(11)/113604/8/$30.00 28, 113604-1 Published by AIP Publishing 113604-2 A Constantin Phys Fluids 28, 113604 (2016) wave crest while the minimal ones are typical for the region below the wave trough Intuitively, one may argue that water particles accelerate downwards under the wave crest, so that a downward dynamic pressure gradient is required On the other hand, under the wave trough particles accelerate upward, resulting in an upward dynamic pressure gradient, while just beneath the surface, about halfway between the crest and the trough, the water will accelerate horizontally, so no vertical dynamical gradient develops and the vertical pressure distribution is practically hydrostatic Using maximum principles for elliptic partial differential equations, we will show that, independent of the wave amplitude of a regular wave train, the maximum of the dynamical pressure occurs at the wave crest and the minimum at the wave trough This result has an interesting consequence: in view of the dynamic boundary condition, which asserts that at the free surface the pressure must be equal to the (constant) atmospheric pressure, we deduce that the variation of the dynamic pressure (the difference between its maximum and minimum values) in a regular wave train is always equal to the wave height times the specific weight of water (density times the acceleration due to gravity) This conclusion is of relevance in the investigation of wave loads on maritime structures since the effect of the wave-induced forces on offshore maritime structures is encoded in the dynamic pressure II PRELIMINARIES Let us present the governing equations for regular irrotational wave trains propagating at the surface of water over a flat bed Compelling practical grounds for the study of this type of water flow are the waves propagating in a canal (at the surface of water of relatively constant average depth), as well as surface waves in sea regions with a flat bed — abyssal plains cover about a third of the Earth’s surface (about as much as all the exposed land combined) We assume the density ρ of the water to be constant, we neglect internal and boundary dissipation so that the flow is inviscid, and we model the influence exerted by the air above the water’s surface as (constant) atmospheric pressure acting on the surface, thus neglecting capillary effects The evolution of the waves is governed by the balance between the restoring force of gravity and the inertia of the system, that is, gravity is the only external force of significance All these assumptions are reasonable in the study of water waves of large amplitude.5 To describe a two-dimensional water wave it suffices to consider a cross section of the flow in the direction of wave propagation since the motion is identical in any plane parallel to it Choose Cartesian coordinates (x, y) with the x-axis pointing in the direction of wave propagation and the y-axis pointing vertically upwards, while the origin is located on the mean water level y = (see Fig 1) Let u(x, y,t), v(x, y,t)) be the velocity field of the two-dimensional flow propagating in the x-direction over the flat bed y = −d, where d > is the mean depth, and let y = η(x,t) be the water’s free surface with mean water level y = Throughout the fluid the equation of mass conservation is ux + vy = (1) and the equation of motion is Euler’s equation ut + uu x + vu y = − Px , ρ vt + uv x + vv y = − Py − g, ρ (2) where P(x, y,t) is the pressure, g is the (constant) acceleration of gravity, and ρ is the (constant) density The boundary conditions associated with (1) and (2) are v=0 on y = −d, (3) on the flat bed and v = η t + uη x on y = η(x,t), (4) 113604-3 A Constantin Phys Fluids 28, 113604 (2016) FIG The profile of regular wave trains is asymmetrical with respect to the mean level, with sharp crests and flat troughs, while linear theory predicts sinusoidal wave-train patterns as well as P = Patm on y = η(x,t), (5) on the free surface The dynamic boundary condition (5), in which Patm represents the (constant) atmospheric pressure at the surface, decouples the motion of the water from that of the air above it, while the kinematic boundary conditions (3) and (4) reflect the fact that both boundaries are interfaces: particles on these boundaries are confined to them at all times.5 Since we are considering surface waves away from their generation area and entering a region of still water, the assumption of an irrotational flow is realistic, that is, the additional specification that the vorticity vanishes, u y − vx = (6) holds throughout the flow Note that the constraint (6) by itself does not guarantee the absence of underlying currents, being compatible with the existence of uniform currents.11 A regular wave train is a smooth travelling wave solution to the governing equations (1)-(6): a smooth solution — it turns out that, in this setting, smoothness forces real-analyticity8 — for which there exists a wavelength L > and a wave speed c > such that the free surface profile η, the fluid velocity (u, v) and the pressure P have period L in the x variable, η depends only on (x − ct), while u, v, and P depend only on (x − ct) and y; note that c > given that we are considering a wave train which propagates in the positive x-direction Moreover, there is a single crest and trough per period, the wave profile is strictly monotone between successive crests and troughs (thus excluding flows with a flat surface), and η, u and P are symmetric while v is antisymmetric about the crest line — by a crest line (trough line), we mean the vertical line directly below a crest (trough) The existence of such waves is well established and photographs of such wave patterns are available.5 III MAIN RESULT We are concerned with the issue of locating in a regular wave train the extrema of the dynamic pressure p(x, y,t) = P(x, y,t) − (Patm − ρg y), (7) defined as the difference between the total pressure P(x, y,t) and the hydrostatic pressure (Patm − ρg y) Using maximum principles for elliptic partial differential equations, we will prove the following result 113604-4 A Constantin Phys Fluids 28, 113604 (2016) Theorem The dynamic pressure in an irrotational regular wave train attains its maximum value at the wave crest and its minimum value at the wave trough, if there are no underlying currents Proof Passing to the moving frame X = x − ct, Y = y, (8) the governing equations become (u − c)u X + vuY = − PX for − d ≤ Y ≤ η(X), ρ (u − c)v X + vvY = − PY − g for − d ≤ Y ≤ η(X), ρ u + v = for − d ≤ Y ≤ η(X), Y X uY = v X for − d ≤ Y ≤ η(X), v = on Y = −d, v = (u − c)η X on Y = η(X), P = Patm on Y = η(X) (9) A reduction of the number of unknowns can be achieved by introducing the stream function ψ(X,Y ), defined up to an additive dimensional constant by ψ X = −v, ψY = u − c (10) Note that the fifth and sixth relations in (9) ensure that ψ must be constant on the flat bed Y = −d and on the free surface Y = η(X) We therefore determine ψ uniquely by setting ψ = on the free surface The governing equations (9) are transformed to the equivalent system ψ X X + ψYY = for − d ≤ Y ≤ η(X), ψ = on Y = η(X), ψ = m on Y = −d, ψ + ψY2 X + Y + d = Q on Y = η(X), 2g (11) for some (dimensional) constants m and Q whose relevance will be elucidated in greater detail below The periodicity of the travelling wave entails an L-periodic dependence of ψ(X,Y ) on the X-variable, a feature that can be verified by inspection of the defining formula Y ψ(X,Y ) = m + [u(X, Y) − c] dY, −d ≤ Y ≤ η(X) −d Here m is the relative mass flux (relative to the uniform flow with constant speed c), defined by η(X ) m= [c − u(X,Y )] dY for all values of X, −d and reflecting the fact that in the moving frame the amount of water passing any vertical line is constant We point out that ψ increases as we descend in the fluid, more precisely ψY = u − c < (12) throughout {(X,Y ) : − d ≤ Y ≤ η(X)} Indeed, at every horizontal level Y below the wave trough we have L L u(X,Y ) dX = u(X, −d) dX (13) L L since the periodic dependence of the harmonic function ψ on the X-variable yields 113604-5 A Constantin ∂Y Phys Fluids 28, 113604 (2016) (1 L L u(X,Y ) dX = L L ) =− ψYY (X,Y ) dX L L ψ X X (X,Y ) dX = 0 We can interpret the uniformity of the mean flow beneath the waves, expressed by (13), as providing us with the formula L u(X, −d) dX κ= L for the strength κ of the uniform current beneath the surface waves The absence of underlying currents — a setting which corresponds to swell entering a region of still water — is therefore captured by the constraint L u(X, −d) dX = 0; (14) note that this means that c is equal to the mean of −ψY along the flat bed, so that the wave speed can be recovered from the wave pattern in the reference frame in which the wave is at rest (this is Stokes’s first definition of the wave speed6) In particular, by the mean-value theorem, the validity of (14) ensures that u must vanish at some location (X0, −d) on the flat bed This enforces (12) since, by the maximum principle, the L-periodicity in the X-variable of the harmonic function ψ(X,Y ) ensures that its maximum and minimum in the domain D = {(X,Y ) : − d ≤ Y ≤ η(X)} will be attained on the boundary For a non-flat free surface the function ψ is definitely not constant throughout D, even though it must be constant along the upper and the lower components of its boundary Thus Hopf’s maximum principle yields that along the upper and lower boundaries of D, we must have ψY 0, the sign being determined by the nature of the extrema that are attained there The fact that ψY (X0, −d) = −c < means that the maximum value is attained on the flat bed Y = −d, and the minimum value must therefore be attained all along the free surface Y = η(X) Thus ψY < on the boundary of D and the maximum principle then ensures (12): ψY < throughout D since ψY is a harmonic function Let us also discuss briefly the relevance of the constant Q in (11) — the total head To elucidate the meaning of this terminology, note that the Euler equation, expressed componentwise by the first two equations in (9), is equivalent to the fact that ψ X2 + ψY2 P − Patm + (Y + d) + =Q 2g ρg (15) throughout {(X,Y ) : − d ≤ Y ≤ η(X)}, for some constant Q Since we are neglecting friction and surface tension effects, the invariance of the left side of (15) is a form of conservation of energy: this is Bernoulli’s law Each term in the expression (ψ X2 + ψY2 )/(2g) + (Y + d), evaluated on the free surface Y = η(X) where P = Patm, has the dimension of length, the first being called the velocity head and representing the elevation needed for the fluid to reach the velocity |∇ψ| during frictionless free fall, and the second term being the elevation head Note that the pressure P is altogether absent from (11) It can be recovered using the dynamic boundary condition and Bernoulli’s law, by setting Ψ2X + ΨY2 , (16) so that Euler’s equation becomes simply an alternative way to express the gradient of P For our purposes, it is convenient to note that the dynamic pressure takes the form P = Pat m + ρgQ − ρg(Y + d) − ρ p = ρg(Q − d) − ρ Ψ2X + ΨY2 (17) throughout the fluid Let us first elucidate the qualitative behaviour of the dynamic pressure along the boundary of the fluid region delimited by a crest and a successive trough, in the moving frame Performing a shift, if necessary, we may assume that the crest is located at X = and the trough at X = L/2 Due 113604-6 A Constantin Phys Fluids 28, 113604 (2016) to (3) and the anti-symmetry of v, we have that v = along the lower and lateral boundaries of the domain D+ = {(X,Y ) : < X < L/2, − d < Y < η(X)} Along the upper boundary we have η X ≤ 0, so that the sixth equation in (9) and (12) ensure v ≥ along this portion of the surface wave profile Since v is harmonic in D+ and v ≡ is not an option — it would mean that ψ depends only on the Y -variable and its constant value along the flat bed would force it to be constant throughout the entire fluid, which is impossible in the presence of waves — the maximum principle ensures that v > throughout D+ But then Hopf’s maximum principle yields vY (X, −d) > for X ∈ (0, L/2), v X (0,Y ) > for Y ∈ (−d, η(0)), v X (L/2,Y ) < for Y ∈ (−d, η(L/2)) (18) (19) (20) While the inequalities (18)-(20) are not new — for example, they were instrumental in the investigation of particle trajectories4,13 — it appears that thus far their relevance for the behaviour of the dynamic pressure was not explored Combined with (12) and (17), (18)-(20) ensure that p is strictly decreasing as we descend vertically in the fluid below the crest, it continues to decrease along the portion {(X, −d) : < X < L/2} of the flat bed as X increases, and this strict decrease persists as we ascend vertically from the bed towards the surface, below the trough (see Fig 2) On the other hand, p is also strictly decreasing as we descend from the crest towards the trough along the upper boundary of D+, in view of (7), (5) and the monotonicity of the wave profile between the crest and a successive trough To investigate whether an extrema of p can occur in the interior of the domain D+, note that the fact that ψ is harmonic yields by means of a direct calculation starting from (17) that p X X + pYY = − 2(p2X + pY2 ) ρ(ψ X2 + ψY2 ) throughout the region −d < Y < η(X); we point out that ψ X2 + ψY2 > by (12) Writing the above quasilinear elliptic partial differential equation in the form of the linear elliptic partial differential equation p X X + pYY = α p X + β pY (21) with variable coefficients α(X,Y ) = − 2p X , ρ(ψ X2 + ψY2 ) β(X,Y ) = − 2pY , ρ(ψ X2 + ψY2 ) FIG Monotonicity properties of the dynamic pressure along the boundary of the fluid region delimited by a crest and a successive trough, in the moving frame: the arrows indicate the direction of descent, pointing from higher values towards lower values 113604-7 A Constantin Phys Fluids 28, 113604 (2016) the maximum principle ensures that, unless p is constant, the extrema of p can only be attained on the boundary of D+ In light of the previously elucidated behaviour of p along the boundary of D+, the claim made in the theorem follows The result proved above has an interesting application to the problem of estimating wave heights from pressure data at the bed Indeed, along the flat bed Y = −d we have, due to (7), that p(X, −d) = P(X, −d) − Patm − ρgd and therefore P(0, −d) − P(L/2, −d) = p(0, −d) − p(L/2, −d) < p(0, η(0)) − p(L/2, η(L/2)) = ρg[η(0) − η(L/2)] since p(X, η(X)) = ρg η(X) by (7) and (5) This yields the estimate H> P(0, −d) − P(L/2, −d) ρg (22) for the wave height H = η(0) − η(L/2); we thus recover a recent result.7 In this context, we recall11 that the pressure in the fluid is maximal at (0, −d), while P(L/2, −d) is the minimum value of the pressure along the flat bed Y = −d, and the average value of the pressure on the bed equals Patm + ρgd IV DISCUSSION AND CONCLUSIONS The behaviour of the dynamic pressure beneath a surface wave is a classic research topic.27 Significant results, due to Longuet-Higgins,18,19 concern the behaviour of averages of spatially periodic motions; in the setting of the present paper the main observation is that, defining the average of the dynamic pressure at a horizontal level Y below the wave trough by L ⟦p⟧(Y ) = p(X,Y ) dX, L using integration by parts, the harmonicity of ψ, (17), and the L-periodic dependence of ψ(X,Y ) on the X-variable yield 2ρ L ∂Y ⟦p⟧ = ψY ψ X X dX, (23) L 2ρ L 2 ) dX ≤ (24) ∂Y2 ⟦p⟧ = − (ψYY + ψ XY L Consequently ∂Y ⟦p⟧ = on the flat bed Y = −d, due to the third relation in (11), and (24) then ensures that the average dynamic pressure is an increasing function of increasing depth This conclusion appears still to represent the state-of-the-art for surface waves The main result of the present paper is that the pointwise dynamic pressure attains its maxima/minima at the wave crest/trough, irrespective of the wave amplitude We see that the pointwise dynamic pressure has a behaviour that is hard to predict from the properties of the average: the effect of the wave field can be subtle and (at first glance) counter-intuitive, with significant cumulative deviations from the properties of the mean flow The following point can thus be appreciated: while the decomposition of a fluid flow into a mean part and a disturbance from that mean is valuable both conceptually and as a framework for performing practical calculations, over-confidence in this approach can sometimes conceal specific nonlinear physical phenomena Since the validity of the main result was established with no formal assumption that limits the wave amplitude, the conclusions apply to waves of small and large amplitude alike; physical reality will, of course, dictate practical limitations on the wave height While the occurrence of the maxima/minima of the dynamic pressure along the free surface is predicted by linear theory (i.e., by the leading-order solution of the Stokes-wave expansion5), the fact that a definite statement holds with 113604-8 A Constantin Phys Fluids 28, 113604 (2016) no restrictions on the wave amplitude is of great practical relevance Earlier attempts to explore the pointwise behaviour of the dynamic pressure were confined to linear or weakly nonlinear regimes1,24 to which the assumption of small amplitude wave motion is inherent We point out that the regular nature of the wave pattern enabled us to locate the global extrema (i.e., relative to the entire fluid domain), whereas for a general surface wave pattern only local extrema appear to be accessible Concerning related research developments, we draw attention to the fact that the type of motion considered in the present paper is purely irrotational flow with no underlying currents Wave-current interactions may alter considerably the behaviour of the dynamic pressure, particularly non-uniform currents (vorticity being their hallmark) can be expected to have important effects, at present under investigation The approach used in the present paper relies heavily on the irrotational nature of the flow, and cannot be applied to deal with flows having non-zero vorticity Nevertheless, provided that no flow-reversal occurs, numerical simulations9,10,15,16,26 and some analytical investigations14 indicate that we should expect qualitatively somewhat similar results 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Vasan, V., Deconinck, B., and Henderson, D., “Recovering the water-wave profile from pressure measurements,” SIAM J Appl Math 72, 897–918 (2012) 23 Peregrine, D H., “Long waves on a beach,” J Fluid Mech 27, 815–827 (1967) 24 Pinto, F T and Neves, A C V., “Second order analysis of dynamic pressure profiles, using measured horizontal wave flow velocity component,” in Proceedings 6th International Conference on Computer Modelling Experimental Measurements of Seas and Coastal Regions, Environmental Studies Series Vol 9, edited by Brebbia, C A., Almorza, D., and Lopez-Aguayo, F (WIT Press, 2003), pp 237–252 25 Umeyama, M., “Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry,” Philos Trans R Soc., A 370, 1687–1702 (2012) 26 Vasan, V and Oliveras, K L., “Pressure beneath a traveling wave with constant vorticity,” Discrete Contin Dyn Syst 34, 3219–3239 (2014) 27 Wehausen, J V and Laitone, E V., “Surface waves,” in Handbuch der Physik (Springer-Verlag, Berlin, 1960), Vol 9, pp 446–778, Part ... to the weight of the fluid A detailed understanding of the behaviour of the dynamic pressure in a water -wave flow is not merely of academic interest since a better pressure field knowledge in wave. .. following result 113604-4 A Constantin Phys Fluids 28, 113604 (2016) Theorem The dynamic pressure in an irrotational regular wave train attains its maximum value at the wave crest and its minimum... represent the state -of- the- art for surface waves The main result of the present paper is that the pointwise dynamic pressure attains its maxima/minima at the wave crest/trough, irrespective of the wave