In section 2.2 we performed the MMS to O(3). Comparing to wavetank experiments, the NLSE often does not give a good agreement with simulations. A higher degree of accuracy can be obtained by performing the MMS expansion on eq. (2.2.14) to the fourth order in steepness, yielding the Higher-order NLSE (HONLSE), Modified-NLSE (MNLS) or Dysthe equation [48]
∂a
where ¯φis the potential mean flow. The potential mean flow term in (10) can be replaced by a Hilbert transform term [49]. This equation has four nonlinear higher-order terms labeled
‘Dysthe correction’. Using the spacelike coordinate transformation, eq. (2.2.33), yields5
∂a
Let us use adimensional coordinates by making the coordinate transformation :
τ=t0/T0, T0=1/(ω0) a0(ξ, τ)= a/A0 (2.4.3)
ξ= x0/L0, L0=1/(22k0) (2.4.4)
(2.4.5) dropping the apostrophe foraleads to
∂a
A comparison between the behavior of the Dysthe equation and the NLSE is exemplified in fig. 2.15 (for a similar comparison see [50]). The first column shows the evolution inξ of the envelope amplitude. The second column shows the evolution of the three main modes: the
5Note that the higher-order dispersion term disappears. Another way to see this is through the dispersion expansion, performed in section 3.2.
carrier modeη0and the upperη+1and lowerη−1sidebands. The third column shows the phase diagram, which will be explained in more detail in section 4.3. For now it suffices to know thatψrepresents the phase of the envelope, andη0a normalization of the energy fraction in the sidebands.
The initial condition for the simulations is a plane wave with equal sideband perturbations (as in fig. 2.9) where the energy fraction in the sidebands is 10−4of the total energy, and the initial carrier wave steepness = 0.12 and the relative phase π. The perturbation wavelength of the sidebands is such that they are at maximum MI.
Row (a) plots the evolution as given by the NLSE. As expected based on the MI, the plane wave with its slight perturbation is unstable. The spectral evolution (center) shows that sidebands grow rapidly, and the main mode η0 decays, coinciding with an increase of the envelope amplitude (left). This behavior occurs in a cyclical fashion, where the initial condition is recovered. The modulation-demodulation cycle is termed the Fermi-Pasta-Ulam recurrence [51].
Row (b) shows the evolution predicted by the Dysthe equation. From the envelope evolution (left), it stands out that the Dysthe terms significantly delay the group velocity, indicated by the angle of the orange dashed lines. In addition, the Dysthe terms delays the recurrence pattern.
The spectral evolution (middle) displays an asymmetric growth of the sidebands. The upper sidebandη+1grows faster than the lower sidebandη−1. These types of asymmetries are created by odd derivatives in the equation. This can easily be seen when looking at the equation in Fourier space, wherei∂∂tA →ωA. The lower the steepness of the carrier wave, the smaller the influence of the nonlinear terms, and the more alike the simulations of Dysthe and NLSE will be.The subsequent rows indicate the influence of the individual terms in the Dysthe correction defined in eq. (2.4.6). TermDIcauses the slowing of the group velocity, and has an asymmetric evolution of the sidebands, but does not affect the recurrence period. TermDI Igives a strong asymmetric evolution of the sidebands. TermDI I Iis the mean flow term. As it is responsible for shrinking the MI range with increasing steepness [49], it decreases the effect of the nonlinearity, lowering the maximum amplitude of the envelope at the focal point. Additionally, in this specific case, it induces as phase-shift for the envelope. In the phase diagram (right) the path the solution traces out crosses zero, and the the undergoes aπphase-shift. That is, the recurrence pattern is shifted inτfrom one recurrence period to the other, indicated by the dashed in the real space.
The recurrence period of the Dysthe equation is determined by the combined effect DI Iand DI I I.
Figure 2.15:NLS/Dysthe simulations performed for illustration.Left: Envelope evolution as a function of the propagation coordinateξ and the transverse time-like coordinateτ. Center: Evolution of the spectral mode: the carrier wave frequency (solid line) the upper sideband (dashed line) and the lower sideband (dotted line).Right: Phase diagram
Viscous damping of water waves
In the previous chapter, water waves were considered to be conservative. They did not lose or gain any energy. In the basic equations for the water wave problem in section 2.2, the water was treated as inviscid and irrotational. However, while not as viscous as honey (dynamic viscosity µ ≈10 Ns/m2), water does have a finite viscosity (µ≈ 0.001 Ns/m2), which, amongst other things, is responsible for the damping of the water waves and is the origin of the vortical flow.
Viscosity is a measure of the resistance to shear stress of a fluid. One can imagine that stirring a pot of honey is much harder than stirring a cup of water, while the densities of the fluids are comparable (1.4 vs 1 kg/l). When moving two plates with a fluid in between, as in fig. 3.1, there is a no-slip boundary condition on the surface of the plates. Here, the fluid will have the same velocity as the plate. In between the plates, the proportionality constant of the gradient of the velocity is the definition of the dynamic viscosity µ: FA = µ∂u∂z, whereFis the force on the top plate, Athe area of the plates and ∂u∂z is the rate of shear deformation, or the shear velocity. The kinematic viscosity is defined asν= µ/ρ.
z x
u
U0 moving
stationary
Figure 3.1: Definition of viscosity: The top plate moves with a velocityU0. Therefore, the fluid must have a velocity 0 at the bottom plate boundary, and a velocityU0at the top plate. Given the force needed to provide such a velocityU0for the top plate, the constant of proportionality with the velocity gradient
∂u
∂z is the dynamic viscosityµ. For Newtonian fluids the velocity profile is linear.
This chapter includes two papers in which we investigate the consequences of the viscous nature of water waves. The first paper, ‘Reconciling different formulations of viscous water waves and their mass conservation’ submitted to Physics Letters A[6], discussed in section 3.1, contrasts and compares three main ways of incorporating the viscosity into the model equations and highlights the equivalence of the vortical boundary layer to a rotational pressure. The second paper, ‘Viscous damping of gravity-capillary waves: Dispersion relations and nonlinear corrections’ published in Physical Review E [4], discussed in section 3.2, considers the influence of viscosity on the dispersion relation.
3.1 Viscous model equations
Generally, the physics of hydrodynamic problems is determined by the boundary conditions.
When water is treated as a viscous fluid, the Cauchy stress tensorσat the boundaries becomes important. The double underline indicates a tensor. The stress tensor completely defines the
Figure 3.2: a) Components of the stress tensor in cartesian coordinates. b) σn is composed of the hydrostatic pressurePand the normal deviatioric stressτn. The tangential component is composed of the tangential deviatoric stressτs. Due to symmetry, only two sides of the square need to be defined.
state of stress at a point inside the fluid. It relates the stress vectorT® to a surface. Let the unit vector ˆedefine the surface that it is normal to, and T®(e)ˆ the stress vector across this surface.
Then,T®(e)ˆ = σ·eˆ, see fig. 3.2a. σ is the addition of the volumetric stress and the deviatoric stress: σ=−PI+τ. For Newtonian fluidsτ= µ(∇ ®u+∇ ®uT), whereT indicates the transpose operator. So we have:
σ=
−P 0
0 −P
+µ
ux wx
uz wz
+µ
ux uz wx wz
=
2µux−P µ(wx+uz) µ(uz +wx) 2µwz −P
=
σx x σxz σz x σzz
(3.1.1) subscripts refer to derivatives and superscripts to the component of the vector. If we want to look at the face of the square fluid parcel (cubic in 3D) close to the boundary, it is more convenient to use normal and tangential components:
σ = µ
2µuss−P µ(uns +usn) µ(usn+uns) 2µunn−P
(3.1.2) The pressure Pis not merely due to the weight of the water, but also to the fluid motion itself. For a steady flow, i.e. not subject to any net-forces or stress, the stress tensor must have only diagonal terms, and this should hold in any coordinate frame. The only tensor that is diagonal in all frames is one in which each diagonal element has the same value. That value is thedefinitionof the static pressureP(Pascal’s law). When there are net-forces or stresses present, the deviatoric stress tensorτis the part of the stress tensor that is left after subtracting P, and is what will cause deformation to the parcel. As the fluid parcel is infinitesimally small, by symmetry, the stress tensor only needs to be defined on two sides.
The normal and tangential component of the stress tensor, i.e. the normal stressσnand the shear stressσs, can be obtained as follows:
σn =nˆ·σ·nˆT =τn−Pn =2µunn−P (3.1.3) The continuity of normal stress is the basis of the dynamic boundary condition: σwatern = σairn. Recall that for a inviscid fluid in section 2.2.1, this reduced to the continuity of pressure:
Pwater= Pair.
For the shear stress we can write:
σs = sˆ·σ·nˆT =τs = µ(uns +uns) (3.1.4)
The shear stress also should be continuous across the boundary: σwaters = σairs . As the density of the air is negligible, the shear stress on the water particle at the surface has to vanish.
A curved surface with a vanishing shear stress implies, to first order, a rigid rotation [52], which can only be performed by a rotational flow (fig. 3.3,right), as opposed to a circular translation (fig. 3.3,left)
rotational irrotational
Figure 3.3:Left:Irrotational movement. Right:Rotational movement.
Therefore, to comply with the boundary condition of a vanishing shear stress for a viscous fluid with a curved boundary, the viscous fluid has to be rotational. In the particular problem of the curved free surface, viscosity and rotationality are therefore linked. Consequently, for a rotational fluid, the velocity vector can no longer simply be written as a scalar potential∇φ=u®, but a rotational part has to be added in what is termed the Helmholtz decomposition:
®
u=∇φ+U® =∇φ+∇ × ®A (3.1.5)
The following paper [6] outlines that by inserting the Helmholtz decomposition into the momentum equation, the Navier-Stokes equation, the problem becomes quite complex. There is now one more unknown to solve for and finding a solution is no longer straightforward. There are different routes to bring this task to a successful ending. We contrast and compare three main approaches to do so. In addition, there is a belief that conservation of mass might be lost in a rotational potential flow-like system. We therefore verify to what extent the three systems obey mass conservation and show that it is exactly the rotational part of the boundary condition that is needed to guarantee that the mass is conserved.
and their mass conservation
D. Eeltink, A. Armaroli, M. Brunetti, J. Kasparian
Group of Applied Physics and Institute for Environmental Sciences, University of Geneva, Switzerland
Abstract
The viscosity of water induces a vorticity near the free surface boundary. The resulting rotational component of the fluid velocity vector greatly complicates the water wave system. Several approaches to close this system have been proposed. We first address how these correspond to each other. Our analysis compares three common sets of model equations. The first set has a rotational kinematic boundary condition at the surface. In the second set, a gauge choice for the velocity vector is made that cancels the rotational contribution in the kinematic boundary condition, at the cost of rotational velocity in the bulk and a rotational pressure. The third set circumvents the problem by explicitly splitting the problem into two domains: the irrotational bulk and the vortical boundary layer. This comparison exemplifies the correspondence between rotational pressure on the surface and vorticity in the boundary layer, and shows where approximations have been used. Second, we disprove the speculation that mass is not conserved in a rotational model, pointing out the physical error of comparing term-by-term the mass conservation of a viscous model to that of an irrotational model.
Keywords: Vorticity, Viscosity, Gravity surface waves, Mass conservation
1. Introduction
For many water wave problems, the Navier-Stokes (NS) equation can be simplified by considering the fluid as in-viscid and incompressible. The continuity equation com-bined with the boundary conditions for the free surface and the rigid bottom forms the water wave problem for the surface elevationηand the velocity potentialφ. This system can be used as the starting point to obtain Nonlin-ear Schr¨odinger (NLS) equation-like propagation models, or can serve as a basis for higher order spectral methods [1, 2, 3].
While this approach is sufficient in many situations, in reality, water is a viscous medium. The viscosity accounts for the obvious fact of wave damping but also plays a more intricate role for instance in downshifting of the spectrum [4], or in stabilizing the Benjamin-Feir instability [5]. In domains such as the dissipation of swells [6], or visco-elastic waves propagating in ice, including viscosity in a set of equations is important [7].
We feel that it is important to get the correct physical in-terpretation of the viscous free surface problem, especially in view of recent interest from the mathematical commu-nity which take these as a starting point. For instance in looking at the well-posedness of the DDZ system [8], or by taking these viscous water wave systems as a basis for new asymptotic models [9].
∗Corresponding author
Email address: jerome.kasparian@unige.ch(J. Kasparian)
The inclusion of viscosity in the water wave problem has been proposed in several ways. One option, hereafter de-noted as System A, used in Ruvinsky et al. [10], Dias et al.
[11] (hereafter R&F and DDZ respectively), includes the rotational part of the velocity vectorU~ into the kinematic boundary condition (KBC) at the surface. The model equations can be simplified by neglecting small terms, based on the premise that the vorticity is confined to a narrow region.
The second option, denoted System B (Dommermuth [12]), makes a gauge choice for the velocity vector such that the rotational contribution in the KBC disappears.
The cost is that this introduces a rotational part of the velocity in the bulk. Moreover, the pressure is split into rotational and irrotational parts too, introducing an addi-tional equation for the rotaaddi-tional pressure that couples to the Navier-Stokes equation. This system is however fully nonlinear and makes no boundary layer approximations.
The third option, denoted System C (Longuet-Higgins [13, 14]), circumvents the difficulty of the rotational kine-matic boundary condition by explicitly splitting the prob-lem into two domains: the irrotational bulk and the vor-tical boundary layer. The former is shown to receive an additional pressure due to the weight of the latter.
Naturally, without approximations, the models describe exactly same physical reality, and have the same solutions.
However, in order to close systems A and C, linearization is applied in certain steps. This approximation can be the source of discrepancies between the models. For instance,
Preprint submitted to - October 16, 2019
δ1
Figure 1: The water wave problem and its relevant length scales.
The velocity potentialφis indicated by the color-scale. The arrows indicate the velocity vector~u, following the gradient ofφ. The sur-face elevation η(x, z) is indicated by the black line. The vortical boundary layerδis indicated by the dashed line for aδ = δk >1, and by the dotted line for aδ=δk <1.
Ref. [15] discusses nonphysical modes that appear after the linear stability analysis of the R&F model. Here, we compare systems A,B. and C and discuss their physical interpretation and equivalence.
Furthermore, the addition of a rotational part to the kinematic boundary condition (KBC), has been claimed to jeopardize the conservation of mass. When verifying the mass conservation, a term-by-term comparison to the irrotational case is often made, as for instance in [8]. In-deed, compared to the irrotational case, an unaccounted rotational term appears. However, we detail that for a curved free surface, the irrotational model is nonphysical, and that it is fact the rotational part that is necessary to conserve the mass. As such, we show that the conserva-tion of mass is not compromised for a rotaconserva-tional system, provided the full nonlinear KBC is used.
The paper is organized as follows: in Section 2 we recall how the kinematic and dynamic boundary conditions are obtained for the water wave problem and how viscosity affects them. In Section 3 we define the three systems of equations and we show how they correspond. In Section 4, we address the mass conservation issue. In Section 5 we interpret the results.
2. Boundary Conditions
The physics of the water wave problem is defined by the boundary conditions. Figure 1 depicts the 2D water wave problem and its relevant quantities and length scales.
The surface elevationη(x, t) is denoted by the black line, the arrows are the local velocity vectors and the color scale refers to the value of the velocity potentialφ(x, z, t), based on a linear wave [16].
2.1. Shear stress: viscosity and vorticity
For non-viscous waves the velocity vector ~u is irrota-tional and can be written as the gradient of a potential
However, in the presence of viscosity, the continuity of tangential stresses at the free surface can only be fulfilled by rotational motion of the fluid [17, 18].
The stress tensor in 2D tangential and normal compo-nents can be written as:
σ=2µuss−P µ(uns+usn) µ(usn+uns) 2µunn−P
(2.2)
whereµis the dynamic viscosity, andPthe pressure. Here and in the following, subscripts denote the partial deriva-tives and superscripts denote the components of a vector, wheresdenotes tangential andnnormal. The tangential stress component is
σs=τs=µ(usn+uns) (2.3) whereτ is the deviatoric stress tensor.
Sinceµin air is much smaller than in water, the shear stress must vanish at the surface, implying usn = −uns. There is thus no relative distortion to the fluid-particle, which due to the curvature of the interface, results in a rotational flow [17]. See Fig. 2 for an illustration of a rotational and an irrotational flow.
rotational irrotational
Figure 2:In an irrotational flow there can be circular paths for the fluid, but each individual fluid particle does not rotate.
For a rotational flow, the Helmholtz decomposition is used to split the velocity field into an irrotational part
∇φ, and a rotational, solenoidal (∇ ·U~ = 0) part,U~
~u=∇φ+U~ =∇φ+∇ ×A~ (2.4) Using the Helmholtz decomposition requires finding a harmonic function φ that satisfies ∇2φ = 0 and a solenoidal field U~ that satisfies the NS equations [19].
Therefore, certain transfers of irrotational flow from the
∇φ term to the U~ are allowed, keeping Eq. (2.4) valid.
That is, while∇φis irrotational (since the curl of a gradi-ent is always 0),U~ can include an irrotational part on top of its rotational part [12].
This implies that one cannot assume a-priori that∇φ contains the full irrotational velocity potential, so that its value differs for different gauge choices.
2.2. Kinematic boundary condition
A key ingredient for the rest of our discussion is the KBC, which we therefore derive explicitly. The KBC en-sures that fluid particles on the free surface always remain 2
P
System A System B System C
Figure 3: (a) System A: The boundary conditions include the balance of normal stress, the vanishing of the shear stress, and the rotational KBC. The grey shaded area indicates thatUn decays over a characteristic lengthδ. (b) System B: The boundary conditions include the splitting of the pressure for the normal stress, the vanishing of the shear stress, and an irrotational KBC. The grey shaded area indicates that whileUn= 0 at the surface, this is not the case within the bulk. (c) System C: The domain is split into two subdomains: Γ1for the bulk, and Γ2for the vortical boundary layer. The boundary conditions for the irrotational domain Γ1onη∗are the continuity of normal stress, with an added pressure due to the weight of ΓII, and an irrotational KBC. The shear stress does not need to vanish atη∗.
Figure 3: (a) System A: The boundary conditions include the balance of normal stress, the vanishing of the shear stress, and the rotational KBC. The grey shaded area indicates thatUn decays over a characteristic lengthδ. (b) System B: The boundary conditions include the splitting of the pressure for the normal stress, the vanishing of the shear stress, and an irrotational KBC. The grey shaded area indicates that whileUn= 0 at the surface, this is not the case within the bulk. (c) System C: The domain is split into two subdomains: Γ1for the bulk, and Γ2for the vortical boundary layer. The boundary conditions for the irrotational domain Γ1onη∗are the continuity of normal stress, with an added pressure due to the weight of ΓII, and an irrotational KBC. The shear stress does not need to vanish atη∗.