Transient integral boundary layer method to calculate the ...

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BioMedical Engineering OnLine 2006, 5:42http://www.biomedical-engineering-online.com/content/5/1/42A δ ∗A δ ∗∗A θwhereas the friction coefficient for a parabolic profile is K ν= -8 π ν. The corresponding momentum correction coefficientis given byγ + 2χ( xt , ) = ,γ + 1which is 4/3 in the parabolic case. Such factors can beused for the purpose of correlating other variables as wellas for direct calculation of pressure drop. We note that inthe presence of a stenosis the total losses under theassumption of Hagen-Poiseuille flow are underestimated[55]. This comes mainly from underestimating the viscousforces and disregarding the losses caused by flow separationat the diverging end of the stenosis [64,65].Boundary layer derived viscous friction and momentum correctionDeveloping flow conditions in ducts of multiply connectedcross-sections generally make it difficult to use theright friction factor. A variety of cross-sections are discussedin [45]. In those situations the similarity parametersare preferably based on the free stream velocity andthe square root of the cross-sectional flow area as characteristiclength scale i.e. = A . Consequently we definethe Reynolds number Re A=V / V , the Womersleyhave multiplied Re and Sr by a factor ofis multiplied byδ ∗θDisplacement tube Figure 6 thickness and displacement area in a deformedDisplacement thickness and displacement area in adeformed tube. Illustration of the displacement thicknessδ*, the momentum thickness θ and the total displacementthickness δ** in a deformed cross-section. The related areasare the displacement area A δ* (dark grey), the momentumarea A θ (light grey) and the total displacement area A δ** = A δ*+ A θ (light + dark grey) respectively.( 43)number or frequency parameter WoA = W / V2Woand the Strouhal number SrA = = ω. Here ωRe Vis the angular frequency defined as ω = 2πf, with f the basefrequency of pulse wave oscillation. In other words weδ ∗∗1 2π , while Woπ . In the calculations we have giventhe Reynolds number inside the stenosis, Re st , based on .As previously mentioned the surface line of the flat portionof the non-circular duct dominates the circular portionat severe deformations, so that the computation ofviscous forces is based on plane wedge flow. Consequentlythe thickness of the boundary layer is estimated inthe xz-plane. Further we assume that the boundary layerhas constant thickness along the circumference as illustratedin Figure 6. The latter assumption allows a simplederivation of the momentum correction coefficient andthe viscous friction term. Integration over the cross-sectionleads to geometric relations for the areas occupied bythe displacement thickness δ* and the total displacementthickness δ** in that cross-section. They are expressed asA δ* = 2B δ* + π [R 2 d - (R d - δ*) 2 ], (44)A δ** = 2B δ** + π [R 2 d - (R d - δ**)2 ], (45)It is obvious that A d > A δ** and A d > A δ* have to be satisfiedto make sure that the flow is not fully developed. Themomentum correction coefficient can be found by satisfyingmass conservation for the mean flow and the core flowbyV (A - A δ* ) = A u, (46)which can be used together with equation (22) and (45)in the definition for the momentum correction (43), sothat( ) =χ xt ,A ∗∗A νA A V da A A − A − δ21∗∗xδ=A∫=.2 A 2 22−( A − A ) ⎛ A∗∗∗δδ1 − δ ⎞⎜⎝ A ⎟⎠( )( 47 )The uniform inflow profile is identical to χ = 1, while thedeveloping profile reaches its far downstream value of1.39 after the entrance length within less than 4.5% fromthe analytical solution for the parabolic flow profile givenin equation (43). We note that in the linearised system thetotal cross-section A in equation (47) is replaced by A d .According to equation (34) the friction factor built withthe pressure dependent surface line, U p (x, t) isFvUpv B+Rdx, twf .ρ τ 2 π20θ( ) =− =−( )( 48 )Computations in a uniform tube show good agreement tothe friction factor of the parabolic profile given in equation(42). After the entrance length the friction factorcomputed via the boundary layer theory reached its fardownstream value to within 7%. Additionally the Fanningfriction factor Reynolds number product in aPage 10 of 25(page number not for citation purposes)
BioMedical Engineering OnLine 2006, 5:42http://www.biomedical-engineering-online.com/content/5/1/42through the boundary layer: t d = δ 2 /V
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