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Dimensionless Parameters

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Dimensionless Parameters GEFD SUMMER SCHOOL Some basic dimensionless parameters and scales The table summarizes (using standard notation as far as it exists) some of the basic quantities encountered in the lectures, whose order of magnitude is usually the ¯rst consideration when assessing a uid-dynamical situation. In uid dynamics, even more than in other branches of physics, quantities like these have more than one meaning and more than one mode of use. For instance, the nominal length and velocity scales L, U may or may not be true scales in the sense that we can estimate the order of magnitude of @=@x as L¡1, of ru as U=L, of ºr2u as ºU=L2, and so on. In a thin boundary layer with thickness ` and downstream lengthscale L, for instance, we might have u ¢ ru » U 2=L and ºr2u » ºU=`2. Then it is U`2=ºL that needs to be of order unity, not the ordinary Reynolds number* UL=º, if viscous forces are to balance typical accelerations. Again, it is often relevant to consider L and U to be external parameters (such as pipe radius and volume ux/pipe area in the Reynolds experiment), especially when interested in `scaling up' or `scaling down' in the sense of identifying the class of problems that reduce to the same problem when suitably nondimensionalized (e.g. half the pipe radius and twice the volume ux requires four times the viscosity, in order to get the same pipe-ow problem). In ordinary low-Mach-number ows, the timescale quite often » L=U, and this is often tacitly assumed when estimating typical material rates of change D=Dt = @=@t + u ¢¢ r » U=L. Dimensionless parameter Symbol and formula Interpretationy U typical advective acceleration Mach number M = csound waves wave-induced particle accel. U U Froude number External F re = = 1=2 " cext. gravity waves (gH) U U Internal Froude number F ri = = " cint. gravity waves NH Small values of M; F re; F ri imply the possibility of `balanced' or `adjusted' ows that do not self-excite the waves in question (cf. mass on sti® spring moved gently), e.g. nearly-incompressible ow when M is small enough, or layerwise-2D strati¯ed ow (e.g. `Los Angeles smog') when F ri is small enough. (Beware: governing equations are `sti®'.) N = buoyancy frequency of stable strati¯cation: N 2 = g @ ln(potential density)=@z. typical advective acceleration Reynolds number R = Re = UL=º typical viscous force/mass 1=2 ºL ¡ Boundary-layer thickness = Re 1=2L = di®usion length for time L=U µ U ¶ (for ow past obstacle) 3 1=4 Kolmogorov microscale `K = (º =²) (Nominal length scale at which viscous dissipation becomes important in three-dimensional turbulence that is dissipating energy at rate ² per unit mass; ² has dimensions length2 time¡3.) 1=4 1=2 Associated velocity and time scales UK » (º²) , tK » (º=²) 2 [Consistency checks: UK`K=º » 1, and tK » `K=º (viscous di®usion time)] * not Reynold's. After Osborne Reynolds who in a famous experiment (MEM, Lecture 1) showed its relevance to whether pipe ow is laminar or turbulent. y when L; U etc are true scales in a ow with simple structure 1 Dimensionless parameter Symbol and formula Interpretationy 2 2 ¡2 (gradient) Richardson Ri = N =(Uz) As for F ri ; Uz = vertical shear. ¡2 number (= F ri if Uz = U=H) (Note that UzH is, quite often, the relevant velocity scale.) typ. advective rate of change of temp. P¶eclet number P e = UL=· typ. di®usive rate of change of temp. (· = heat di®usivity) momentum di®usivity Prandtl number º=· heat di®usivity momentum di®usivity Schmidt number º=· s solute di®usivity typical relative advective accel. Rossby number Ro = U=­L » U=2­L typical Coriolis accel. (Kibel' number in » U=cinertia waves ¿ ) Soviet literature) (Ro 1 rotationally sti®) typical viscous force/mass Ekman number E = º=­L2 typical Coriolis accel. (For relevance to spindown time ­¡1E¡1=2, L needs to be a scale in the ­ direction) Ekman-layer thickness (º=­)1=2 Di®usion length for scale time ­¡1 Prandtl's ratio H=L » f=N of scales f = 2­ sin(latitude) (Natural vertical-to-horizontal aspect ratio for strati¯ed, rotating ow at low F ri and Ro) Associated quantities: Rossby length L » NH=f (also `Rossby radius'; no standard symbol) Rossby height H » fL=N (no standard symbol) Burger number Bu = N 2H2=f 2L2 (sometimes de¯ned the other way up) (H and L are vertical and horizontal length scales; H ¿ L in atmosphere and ocean) 0 3 g H 0 ¢½ Rayleigh number Ra = (g = g ) ºk ½ Ra is the product ReP e for vigorous thermal convection (assuming that the velocity scale U is such that vertical advective acceleration U 2=H » buoyancy acceleration » g0). total vertical heat or buoyancy ux in a thermally convecting layer Nusselt number conductive heat or buoyancy ux if convection suppressed vertical eddy buoyancy ux Flux Richardson number Rif = Uz £ eddy momentum ux Rif arises from the turbulent energy equation for strati¯ed shear ows. It compares the rate at which eddies do work against gravity (in reducing the stable strati¯cation) with the rate at which they acquire energy from the mean shear Uz. y when L; U etc are true scales in a ow with simple structure 2.
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