chap2.tex (9665B)
1 \section{Dimensional Analysis} 2 Our derived model of fluid mechanics yields formal connections between 3 physical quantities. These quantities bear units, e.g. the velocity of fluid 4 particles $\mathbf{u}$ has the ``SI'' unites of $\frac{m}{s}$, meters per 5 second. The idea is to make use of these scales and formulate a model, where 6 the quantities are nondimensionalized, i.e. to get rid of physical units by 7 scaling each quantity appropriately. The appropriate length scales are that 8 of the typical water depth $h_0$ and the typical wavelength $\lambda$ of a 9 surface wave. 10 11 \subsection{Nondimensionalisation\label{sec:nondim}} 12 In summary we use these adaptations for the scales 13 14 \begin{itemize} 15 \item $h_0$ for the typical water depth 16 \item $\lambda$ for the typical wavelength 17 \item $\frac{\lambda}{\sqrt{g h_0}}$ time scale of wave propagation 18 \item $\sqrt{g h_0}$ velocity scale of waves in $(x, y)$ 19 \item $\frac{h_0 \sqrt{g h_0} }{\lambda}$ velocity scale in the $z$ 20 direction. 21 \end{itemize} 22 Additionally we write the 23 boundary condition on the free surface as follows 24 \begin{align} 25 h = h_0 + a \eta (\mathbf{x}_\perp, t) = z, 26 \end{align} 27 where $a$ is the typical amplitude and $\eta$ nondimensional function. All in 28 all we have the following scaling for the physical quantities of our context 29 \begin{align} 30 &x \rightarrow\ \lambda x, \quad u \rightarrow \sqrt{gh_0} u, \\ 31 &y \rightarrow\ \lambda y, \quad v \rightarrow \sqrt{gh_0} v, \qquad 32 t\rightarrow \frac{\lambda}{\sqrt{gh_0}}t,\\ 33 &z \rightarrow\ h_0 z, \quad w \rightarrow 34 \frac{h_0\sqrt{gh_0}}{\lambda} w. 35 \end{align} 36 with 37 \begin{align} 38 h = h_0 + a \eta, \qquad b \rightarrow h_0 b. 39 \end{align} 40 The pressure is also rewritten into 41 \begin{align} 42 P = P_a + \rho g(h_0 -z) + \rho g h_0 p, 43 \end{align} 44 where $P_a$ is the atmospheric pressure, the term $h_0-z$ represent the 45 hydrostatic pressure distribution, i.e. pressure at depth and the term with the pressure 46 variable $p$ measures the deviation from the hydrostatic pressure 47 distribution. We have $p\neq 0 $ for wave propagation. Now we can perform a 48 rescaling of the Euler's Equation of Motion, we introduce the notation 49 \begin{align} 50 &t = \frac{\lambda}{\sqrt{gh_0}}\tau,\quad x = \lambda \xi,\quad u = 51 \sqrt{gh_0} \tilde{u}\\ 52 &y = \lambda \chi,\quad v = \sqrt{gh_0} \tilde{v}\\ 53 &z = h_0 \zeta, \quad w = \frac{h_0\sqrt{gh_0} }{\lambda}\tilde{w}. 54 \end{align} 55 We start off with the $x$ coordinate, substitute and apply the chain rule 56 leaving us with 57 \begin{align} 58 \frac{Du}{Dt} 59 &= \frac{\partial u}{\partial t} +u \frac{\partial 60 u}{\partial x} \\ 61 &= \sqrt{gh_{0}}\frac{\partial \tilde{u}}{\partial \tau} \frac{\partial 62 \tau}{\partial t} +gh_0 \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} 63 \frac{\partial \xi}{\partial x} \\ 64 &= \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau} 65 + \tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right), 66 \end{align} 67 on the other hand 68 \begin{align} 69 \frac{gh_0}{\lambda} \left( \frac{\partial \tilde{u}}{\partial \tau} 70 +\tilde{u} \frac{\partial \tilde{u}}{\partial \xi} \right) 71 &=-\frac{1}{\rho}\frac{1}{\lambda}\frac{\partial P}{\partial x} \\ 72 &=-\frac{ g h_0 }{\lambda}\rho \frac{\partial p}{\partial \xi}. 73 \end{align} 74 Thereby the rescaling evolves to 75 \begin{align} 76 \frac{D \tilde{u}}{D\tau} = -\frac{\partial p}{\partial \xi}. 77 \end{align} 78 Because of the same scaling in $y$ we get the same result as in $x$, that is 79 \begin{align} 80 \frac{D \tilde{v}}{D\tau} = -\frac{\partial p}{\partial \chi}. 81 \end{align} 82 In the $z$ coordinate we have 83 \begin{align} 84 \frac{Dw}{Dt} 85 &= \frac{\partial w}{\partial t} +w \frac{\partial 86 w}{\partial \zeta} \\ 87 &= \frac{h_0\sqrt{gh_0}}{\lambda} \frac{\sqrt{gh_0}}{\lambda} 88 \frac{\partial \tilde{w}}{\partial \tau} + \frac{1}{h_0} 89 \frac{h_0\sqrt{gh_0} }{\lambda} \frac{h_0\sqrt{gh_0}}{\lambda} 90 \tilde{w}\frac{\partial \tilde{v}}{\partial \zeta}\\ 91 &= \frac{h_0^2g}{\lambda}\left( \frac{\partial \tilde{w}}{\partial \tau} 92 + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right) . 93 \end{align} 94 On the other side we have 95 \begin{align} 96 \frac{h_0^2g}{\lambda^2}\left( \frac{\partial \tilde{w}}{\partial \tau} 97 + \tilde{w}\frac{\partial \tilde{w}}{\partial \zeta} \right) 98 &= 99 -\frac{1}{h_0\rho} \frac{\partial P}{\partial z} +g \\ 100 &=-\frac{1}{h_0\rho}\left(-\rho gh_0 \frac{\partial \zeta}{\partial 101 \zeta} + \rho gh_0 102 \frac{\partial p}{\partial \zeta} \right) + g \\ 103 &= -g \frac{\partial p}{\partial \zeta}. 104 \end{align} 105 In total for the $z$ direction we get 106 \begin{align} 107 \underbrace{\left( \frac{h_0}{\lambda} \right)^2}_{=: \delta^2} 108 \frac{Dw}{Dt} = -\frac{\partial p}{\partial \zeta}, 109 \end{align} 110 where $\delta$ is the \textbf{long wavelength} or \textbf{shallowness} 111 parameter, a very important constant for developing model hierarchies. For 112 clarity we resubstitute for $x, y, z, t, u, v$ and $w$, and for completeness 113 we display the equations again, which are 114 \begin{align}\label{eq:nondim-motion} 115 \frac{Du}{Dt} = - \frac{\partial p}{\partial x}&, \quad 116 \frac{Dv}{Dt} = - \frac{\partial p}{\partial y}, \quad 117 \delta^2\frac{Dw}{Dt} = - \frac{\partial p}{\partial z}, \\ 118 &\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} 119 +\frac{\partial w}{\partial z} = 0. 120 \end{align} 121 We can now turn our attention to the boundary conditions, on both free 122 surface $z=h$ and the bottom $z=b$ we have $z \Rightarrow h_0 z$ and thereby 123 \begin{align} 124 z = 1+ 125 \underbrace{\frac{a}{h_0}}_{:=\varepsilon}\eta(\mathbf{x}_\perp,t) \quad 126 \text{and}\quad z= b, 127 \end{align} 128 where we arrive at our second very important parameter $\varepsilon$ called 129 the \textbf{amplitude} parameter. As for the kinematic condition, we 130 substitute the free surface $z=h = 1+\varepsilon \eta$ and get 131 \begin{align} 132 \frac{Dz}{Dt} = \varepsilon\left(\eta_t + (\mathbf{u}_\perp 133 \nabla_\perp)\eta\right) \qquad \text{on}\;\; z= 1+\varepsilon \eta. 134 \end{align} 135 Respectively the bottom condition is not changed 136 \begin{align} 137 w = b_t + (\mathbf{u}_\perp \nabla_\perp) b \quad \text{on}\;\; z= b. 138 \end{align} 139 The general dynamic condition for $h = h(x, y, t)$ yields a rescaling of the 140 curvature in terms of 141 \begin{align} 142 \frac{1}{R} 143 &= \frac{(1+h_y^2)h_{x x} + (1+h_x^2)h_yy - 2h_xh_yh_{xy} 144 }{\left(h_x^2+h_y^2 +1 \right)^{\frac{3}{2}} } \\ 145 &= -\frac{\varepsilon h_0}{\lambda^2} \frac{( 146 1+\varepsilon^2\delta^2\eta_y^2 )\eta_{x x}+ 147 (1+\varepsilon^2\delta^2\eta_x^2)\eta_{yy} - 148 2\varepsilon^2\delta^2\eta_x\eta_y\eta_{xy}}{\left( 149 1+\varepsilon^2\delta^2\eta_x^2+\varepsilon^2\delta^2\eta_y^2 150 \right)^{\frac{3}{2}} }, 151 \end{align} 152 together with the pressure difference 153 \begin{align} 154 \Delta P = \rho g h_0(p - \varepsilon \eta) = \frac{\Gamma}{R}, 155 \end{align} 156 leaving us ultimately with the dynamic condition 157 \begin{align} 158 p-\varepsilon\eta= \varepsilon\left( \frac{\Gamma}{\rho g\lambda^2} 159 \right) \left(\frac{\lambda^2}{\varepsilon h_0}\frac{1}{R}\right), 160 \end{align} 161 where $W_e = \frac{\Gamma}{\rho g h_0^2}$ is the \textbf{Weber number}. This 162 dimensionless parameter can be considered as a measure of the fluid's inertia 163 compered to its surface tension, which satisfies the relation 164 \begin{align} 165 \delta^2 W_e = \frac{\Gamma}{\rho g \lambda^2}. 166 \end{align} 167 \subsection{Scaling of Variables} 168 Admits a simple observation of the governing equations in the last chapter we 169 notice that $w$ and $p$ on the free surface $z = 1 + \varepsilon\eta$ are 170 directly proportional to $\varepsilon$. Hence we want to ''scale this way`` 171 by introducing the following transformation 172 \begin{align} 173 p \rightarrow \varepsilon p, \quad w \rightarrow \varepsilon w, \quad 174 \mathbf{u}_\perp \rightarrow \varepsilon \mathbf{u}_\perp. 175 \end{align} 176 Because of this scaling our material derivative changes slightly to 177 \begin{align}\label{eq:mod-material} 178 \frac{D}{Dt} = \frac{\partial }{\partial t} + \varepsilon\left(u 179 \frac{\partial }{\partial x} + v \frac{\partial }{\partial y} + w 180 \frac{\partial }{\partial z} \right) 181 \end{align} 182 A simple recalculation yields the rescaled, nondimensionalized Euler's 183 Equation of motion are the same as in equations \ref{eq:nondim-motion} with 184 the modified material derivative from \ref{eq:mod-material}, and the boundary 185 conditions are 186 \begin{align} 187 \begin{drcases} 188 p = \eta - \frac{\delta^2\varepsilon h_0}{\lambda^2} \frac{W_e}{R}\\ 189 w = \frac{1}{\varepsilon}\eta_t + (\mathbf{u}_\perp \nabla_\perp)\eta 190 \end{drcases} \quad 191 \text{on}\;\; z = 1+\varepsilon\eta\\ 192 w =\frac{1}{\varepsilon}b_t + (\mathbf{u}_\perp \nabla_\perp)b \quad 193 \text{on}\;\; z=b 194 \end{align} 195 \subsection{Model Hierarchies} 196 As we have derived a model of fluid mechanics, with small parameters 197 $\varepsilon$ and $\delta$, we can conduct a series of classifications on the 198 long wave problem and perform asymptotic analysis on them. The main 199 hierarchies important in this review are derived from the following problem 200 classifications 201 \begin{itemize} 202 \item $\varepsilon\rightarrow 0$: linearized problem, small amplitude 203 \item $\delta\rightarrow 0$: shallow Water, long-wave 204 \item$\delta \rightarrow 0;\; \varepsilon\approx1$: shallow Water, large 205 amplitude 206 \item $\delta\ll 1;\; \varepsilon\approx\delta$: shallow water, medium 207 amplitude 208 \item $\delta\ll 1;\; \varepsilon\approx\delta^2$: shallow water, small 209 amplitude 210 \item $\delta \gg 1;\; \varepsilon\cdot\delta\ll 1$: deep water, small 211 steepness. 212 \end{itemize} 213 214 215