Turbulent Boundary Layers: Roughness Effects

Transcription

1 Marine Hydrodynamics, Spring 2005 Lecture Marine Hydrodynamics Lecture 19 Turbulent Boundary Layers: Roughness Effects So far, we have assumed a hydraulically smooth surface. In practice, it is rarely so, due to fouling, rust, rivets, etc.... Viscous sublayer U o v = characteristic roughness height To account for roughness we first define an equivalent sand roughness coefficient (units: [L]), a measure of the characteristic roughness height. The parameter that determines the significance of the roughness is the ratio We thus distinguish the following two cases, depending of the value of the ratio on the actual surface - e.g., ship hull. 1. Hydraulically smooth surface For < v <<, where v is the viscous sub-layer thicness, does not affect the turbulent boundary layer significantly. << 1 Cf C f, smooth C f = C f (R el ) 1

2 2. Hydraulically rough surface For >> >> v, the flow will resemble what is setched in the following figure. separation v In terms of sand grains: each sand grain can be thought of as a bluff body. The flow, thus separates downstream of each sand grain. Recalling that drag due to separation = form drag >> viscous drag we can approximate the friction drag as the resultant drag due to the separation behind each sand grain. >> 1 C f C f, rough C f = C f (,R el ) L wea dependence /l C f C D F (Re L ) /l = constant C frough R L C fsmooth C f, rough has only a wea dependence on R el, since for bluff bodies C D = F (R el ) 2

3 In summary The important parameter is /: (x) << 1 : hydraulically smooth >> 1 : rough (x) Therefore, for the same, the smaller the, the more important the roughness Corollaries 1. Exactly scaled models (e.g. hydraulic models of rivers, harbors, etc... ) Same relative roughness: L const for model and prototype ( ) L = L L 1/5 R el For R e model << R e prototype : for ReL ( ) ( ) < m p (C f ) m < (C f ) p 3

4 2. Roughness Allowance. Often, the model is hydraulically smooth while the prototype is rough. In practice, the roughness of the prototype surface is accounted for indirectly. C f U = R = constant ν ΔC f remains constant with R l R L C fsmooth U For the same ship (R e same), different gives different R e =. ν For a given R e, the friction coefficient C f is increased by almost a constant for U = Re = const over a wide range of R el. ν If the model is hydraulically smooth, can we account for the roughness of the prototype? Notice that ΔC f =ΔC f (R e ) has only a wea dependence on R el. We can therefore, run an experiment using hydraulically smooth model, and add ΔC f to the final friction coefficient for the prototype C f (R el )= C f smooth +ΔC f (R }{{ e ) } not (R e L ) Gross estimate: For ships, we typically use ΔC f = R e R e Reality: = ( ) = = 4/5 /L R el R el R 1/5 el as ReL, i.e., ΔC f smaller for larger R el. Hughes Method Adjust for R el dependence of C f rough. i.e., As R el,δc f. C f rough = Cf smooth (1 + γ) = ΔC f = γc f smooth (R e L ) 4

5 Chapter 5 - Model Testing. 5.1 Steady Flow Past General Bodies - In general, C D = C D (R e ). - For bluff bodies Form drag >> Friction drag C D const C P (within a regime) Recall that the form drag (C P ) has only regime dependence on Reynold s number, i.e, its NOT a function of Reynold s number within a regime. - For streamlined bodies C D (R e )= C f (R e )+ C P Steps followed in model testing: (a) Perform an experiment with a smooth model at R em (R em << R es ) and obtain the model drag C DM. (b) Calculate C PM = C DM C fm (R em )= C PS = C P ; C DM measured, C fm (R em ) calculated. (c) Calculate C DS = C P + C fs (R es ) (d) Add ΔC f for roughness if needed. C P measured C f (R m) calculated R m C D predicted C f (R ship) calculated R ship C f (R ship ) R 5

6 Caution: In an experiment, the boundary layer must be in the same regime (i.e., turbulent) as the prototype. Therefore turbulence stimulator(s) must be added. CP turbulent regime TBL LBL $ TBL U MODEL Laminar Cf Turbulent Cf R Turbulent boundary layer to be triggered here Drag on a ship hull For bodies near the free surface, the Froude number F r is important, due to wave effects. Therefore C D = C D (R e,f r ). In general the ra- R e gl tio = 3. It is impossible to easily scale both R e and F r. For example F r ν R e L m 1 ν m g m = constant and = = or = 1000! F r L p 10 ν p g p This maes ship model testing seem unfeasible. Froude s Hypothesis proves to be invaluable for model testing calculate measure indirectly {}}{ C D (R e,f r ) C f (R e ) + C R (F r ) C f for flat plate residual drag of equivalent wetted area In words, Froude s Hypothesis assumes that the drag coefficient consists of two parts, C f that is a nown function of R e, and C R,a residual drag that depends on F r number only and not on R e. Since C f (R e ) C f (R e ) flat plate, we need to run experiments to (indirectly) get C R (F r ). Thus, for ship model testing we require Froude similitude to measure C R (F r ), while C f (R e ) is estimated theoretically. 6

7 5.1.3 OUTLINE OF PROCEDURE FOR FROUDE MODEL TESTING (S SHIP M MODEL ; in general ν S = ν M, and ρ S = ρ M ) 1. Given U S, calculate: F rs = U S / gl S = F rm 2. For Froude similitude, tow model at: U M = F rs gl M 3. Measure total resistance (drag) of model: Measure D M 4. Calculate total drag coefficient for model: C DM = 0.5ρM UM 2 S M wetted area Use ITTC line to calculate C f (R em ): C f (R em )= (log10 R em 2) 2 D M 6. Calculate residual drag of model: C RM = C DM C f (R em ) 7. Froude s Hypothesis: C RM (R em,f r )= C RM (F r )= C RS (F r )= C R (F r ) Use ITTC line to calculate C f (R es ): C f (R es )= (log10 R es 2) 2 9. Calculate total drag coefficient for ship: C DS = C R (F r )+ C f (R es )+ ΔC f = typical value ( ) Calculate the total drag of ship: D S = C DS 0.5ρ S U S S S wetted area 11. Calculate the power for the ship: P S = D S U S 12. Repeat for a series of U S 7