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1 3 Enkeleida Lushi [email protected] Under the supervision of Dr. Mary Catherine Kropinski Department of Mathematics Simon Fraser University Efficient Computational Methods for Water Waves via Boundary Integral Methods 4 Introduction & Purpose • Surface waves have interested a considerable number of mathematicians for many centuries and still intrigue us today. • A lot has been done to study this nonlinear problem, yet not all is known. • We need to understand the underlying mechanisms that influence them. • Why? They are everywhere: sloping waves in the beaches, flood waves in rivers, free oscillations in lakes and harbors, fronts in the atmosphere, tsunami waves, to mention just a few. • Computational Fluid Dynamicists have been recently investigating the myriad of challenging problems that come up from a new point of view. • Why Water Waves? It is an interesting representative of the Free Boundary Problems and quite challenging on itself. 5 Free Boundary Problems in Fluid Dynamics They arise when the dynamics of: • !"# %&’()*+, &- * -.’/) • 0 %&’()*+, %#12##( 12& -.’/)3 • 0 %&’()*+, 2/1"/( 1"# -.’/) /13#.- must be simultaneously determined with the dynamics of the fluid. !"# %&’()*+, ! 4"*(5#3 /( 1/6#7 *3 2#.. *3 1"# -.’/) )&6*/( " 8 ) ( t ! ) ( t " ) ( t u x ) ( t v 6 Examples of Free Boundary Problems • Inertial Flows - Kelvin-Helmholtz instability with surface tension (fig.1 – from H. Ceniceros’ website) - Rayleigh-Taylor instability - Capillary Waves (will see this one later) - Axisymmetric Flow • Stokes’ Flow – Bubbles and drops in viscous flow. • Hele - Shaw Flows - Pattern Formation/ Selection (fig. 2) - Surface Tension driven Singularity Formation - Taylor-Saffman Instability (fig. 2 – from M. Shelley’s website) - Axisymmetric Porous Flow • Materials’ Science ! Why are these problems interesting and difficult? • "#$%& () *+, -$./() ! /0, )$)1#$2/#3 ,454 *+, 2$)*()6(*7 ,86/*($) (& / 5#$9/# 2$)&*0/()* $) *+, :#$%4 • ;+, :#$% -$./() !<*= (& *(.,1-,>,)-,)*3 /)- &$ (& *+, 9$6)-/07 !<*= /)- 2/) 9,2$., ?,07 0/.(:(,- /)- ,?,) &()56#/04 • @60:/2, &*0,&&,& 2/) 9, 2$.>#(2/*,- 97 )$)#(),/0 -,>,)-,)2(,& $) *+, 5,$.,*07 <,454 *+, &60:/2, *,)&($) (& -,>,)-,)* $) *+, 260?/*60,4 0 " # $ % u 8 Equations and Boundary Conditions Navier-Stokes Equations: Boundary Conditions: ! " # " $ # % & ’ ’ j g p u u t u i i i i i ( 1 ) ( 0 $ % # i u 0 ] [ $ % ! ) n u *+ $ ) ] [ p !"#$%&’"( *+#,"’"+# -&./&($01+2#3 *+#,"’"+# For this derivation, consider the motion of a general 2-fluid flow in 2-D which is: - Incompressible - Irrotational - Inviscid - Has Infinite Depth. 4/2", 56 !57 257 .5 "8’9 : 4/2", ;6 !;7 2;7 .; ! The Variables & Equations in the Boundary Integral Formulation !"#$%&’($ *+,-#-+". /0!1#2 3 40!1#25-60!1#2 7+8*9$4 :$9+(-#6. ;0!1#2 3 <0!1#2=->!1#2 7+8*9$4 ?+#$"#-’9. !0!1#2 3 "0!1#2 5-"0!1#2 :+%#$4 @A$$# @#%$"B#A. #0!1#2 7<%>’#<%$. $0!1#2 ' ) 2 ) ' ( ) ( cot( ) ' ( 4 1 2 0 * ! ! ! ! " # # d z z i dz d iv u W dt dz $ % & ’ & % & & () * & + % gy W t 2 | | 2 1 2 / 3 2 2 ) ( ! ! ! !! ! !! ) y x y x x y + % & ' ) 2 ) ' ( ) ( cot( ) ' ( 4 ) ( Re 2 ) ( ) ( 2 0 ! ! ! ! " # ! ! " * # ! ! d z z i z z $ % + & 10 • Numerical simulations using Boundary Integral Methods are sensitive to numerical instabilities. ! A compatibility between the choice of quadrature rule for the singular integral and the discrete derivatives must be satisfied. ! For Spectral Accuracy we choose: - Pseudospectral Approximations for the Space Derivatives - Alternating Trapezoidal Rule for the Singular Integral • Integrator for the ODE-s: 4 th order Adams-Bashforth or Runge-Kutta method. • GMRES algorithm is used to solve iteratively for !. • A 4 th order extrapolation method in time is used to obtain a more accurate initial guess for ! before solving iteratively for it. • Doubling of points is often needed when the wave enters the breaking regime. Numerics & Implementation !! Standing wave without Surface Tension "#$%&’( * $% +,( -./( /$#+(0 1,((+ 1+#(23+, ! 4$ 5,(56 +,.+ .’3$#&+,7 -$#61 5$##(5+’89 . 1+.2:&23 -./( &1 5$7";+(:< =( (0"(5+ +,( &2+(#%.5( "$1&+&$2 2$+ +$ :&%%(# 7;5, .%+(# (.5, "(#&$: >?"@ &2 +,( ’&7&+ +,( .7"’&+;:( $% +,( -./( &1 7;5, 17.’’(# +,.2 +,( :("+,< 12 Standing wave with Surface Tension When surface tension is added into the equation, the computations require a smaller time step, but with filtering, the wave can be advanced for quite a few periods. We notice that the period of the standing wave is slightly shorter than when without surface tension. 13 Breaking wave without Surface Tension To show that the algorithm works well even in a nonlinear regime, a breaking wave is computed well into the breaking time. 14 De-aliasing Issues Beale, Hou, Lowengrub proved in 1996 the method is convergent provided careful de-aliasing is used for the high wave-numbers. The necessity of the filtering of high wave-numbers can be seen from the spectrum plots below: Computations without filtering Computations with filtering !" Problems that Arise • #$%& ()*+,-% .%&(/0& /( ,11%12 .$% -034).,./0&( ,*% (./++5 • 6078/&9 /.%*,./8%7: +0* ! .,;%( ./3% ,( .$% &)3<%* 0+ /.%*,./0&( ;%%4( /&-*%,(/&95 • 6.,</7/.: =0&(.*,/&. /( 0+ .$% +0*3> 2 / 3 ~ ) ( x C t ! " ! # $ # # % min ~ & ?$/( *%(.*/-./0& /( @)/.% (%8%*% +0* .$% ./3%A(.%4 ,&12 <%-,)(% 40/&. -7)(.%*/&9 $,44%&( /& , <*%,;/&9 B,8% ,&1 " -,& 9%. 8%*: (3,775 16 Point Clustering !" The fix: computations in the (!,") frame ) Im( | | 2 1 ) ( 1 1 2 ________ z g U U W dt d U U U dt d U dt d T A A T N N ! " " # " " # ! # $ $ $ $ % & ’ ( % & & % % & #$%&% ($% )%*+,-(. ,+/0+1%1(2 3&%4 ) Im( | | 1 W z z U N $ $ ! ! # ) Re( | | 1 W z z U T $ $ ! # ' ] [ 0 _________ $ % % $ $ $ d U U U U N N A A ) ! " ! # !" Small Scale Decomposition (S.S.D.) #$% ’()*+,+- -%.)/ ,.% -$% 01.2,-1.% ! *+ -$% 3%.+(144* 561,-*(+ ,+’ "7+8"# *+ -$% %2(41-*(+ %61,-*(+ 9(. $: ;- 0,+ <% /$(=+ -$,-> #$%+ -$% %2(41-*(+ %61,-*(+/ ,.% =.*--%+ ,/> ) ( ) ( 2 1 ! ! " R H U N # $ % & ! 2 ’ Q dt d P H dt d # $ # $ % %% ( " ) & & " ( ) ( 1 2 ________ N U dt d % ( " * $ 20 Capillary Waves’ Appearance When surface tension is added in the computations of the breaking wave, we notice capillary waves start to appear at the tip of the breaker. !" !"#$%"%&’(’)*& !,,-%, . #$$% ’( )*+$ *, -,-’-*. /*+$ 01(2-.$ /-’) $34*..5 60*7$% 0(-,’6 ’( 6’*1’ /-’)8 . 9)$ $34*’-(,6 2(1 ! *,% " 7*, :$ -,’$;1*’$% 46-,; <=>? @$’)(%6A 6-,7$ ’)$5 %$7(40.$ ,-7$.5 -, B(41-$1 60*7$ *,% *1$ $*65 ’( -@0.$@$,’8 . C’-.. ,$$% # 2(1 ’)$ +$.(7-’5 DA *,% /$ 6(.+$ 2(1 -’ -’$1*’-+$.5A *;*-,8 >E’1*0(.*’-(, -, ’-@$ )$.068 . F=G>C -6 46$% 2(1 $22-7-$,’.5 7(@04’-,; ’)$ -,’$;1*. $34*’-(,6 2(1 !8 . B*6’ =4.’-0(.$ =$’)(%6 6)(4.% :$ 46$% ’( $+*.4*’$ ’)$ +$.(7-’5A :4’ ’)-6 )*6 ,(’ 5$’ :$$, -,7(10(1*’$% -,’( ’)$ *.;(1-’)@8 21 Benefits of Small Scale Decomposition • Computations are not stiff, the time stepping constraint is not as severe as before. • The points are equi-spaced, hence no point clustering occurs, if Ua is computed accordingly. • Less computational time is needed. • The interface is well-resolved, even in the breaking regime. 22 Summary and Conclusions • The understanding of the movement of water waves and their underlying mechanisms is important to the Mathematics/Engineering Community, but might have an impact in the Industry. • The problem is difficult to derive and challenging to implement numerically. • There are quite a few numerical stability issues, but computations in the equal arc-length frame help levitate some. • The effects of the surface tension on the waves can be noticed. • Capillary waves, resulting from gravity and surface tension effects, can be seen in the equal-arc-length computations of the breaking waves. 23 Future Work • Get a fully working code for the Small Scale Decomposition computations. • Use a higher order semi-implicit scheme. • Computations over a finite-depth topography with different profiles, with and without surface tension, to simulate shallow water waves. • Incorporate weak viscosity in the equations and see what happens. We expect it to dampen the effect of the capillary waves. • Do you have any suggestions? 24 On the funny side, my computations have a long way to go before they catch up with things like this:
