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Model and the Mixing Length Model water Article On the Difference of River Resistance Computation between the k − # Model and the Mixing Length Model Wenhong Dai 1,2,3,*, Mengjiao Ding 1 and Haitong Zhang 1 ID 1 College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China; [email protected] (M.D.); [email protected] (H.Z.) 2 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Nanjing 210098, China 3 National Engineering Research Center of Water Resources Efficient Utilization and Engineering Safety, Hohai University, Nanjing 210098, China * Correspondence: [email protected]; Tel.: +86-134-5189-1670 Received: 22 May 2018; Accepted: 27 June 2018; Published: 29 June 2018 Abstract: River resistance characteristics, which can be reflected by the resistance factor, have an impact on flow and sediment transport. In the classical theory, Prandtl proposed the mixing length model for the simulation of the turbulence, and von Kármán established the logarithmic formula of the flow velocity distribution. Based on that, the expression of the resistance factor can be derived. With the development of the numerical technology, the k − # model has been widely applied in the channels computation. However, for the different closure ways of the Reynolds stress in turbulence equations, the outcomes of the k − # model and the Prandtl mixing length model are not exactly identical. In this paper, both qualitative and quantitative studies are carried out on the difference between these two models, with respect to the resistance factor. This difference is evaluated by the ratio of the resistance factor computed with the two models. The result shows that with the increment of the relative flow depth, the ratio first increases and then decreases. Moreover, it is also affected by the bed slope. Therefore, the difference should be taken into account when a comparison is made between the simulation results of the k − # model and the classical theory of river mechanics. Keywords: resistance k − # model; mixing length model; turbulence; open-channel uniform flow 1. Introduction In the (vertical) two-dimensional (2D) steady uniform open channel flow, as shown in Figure1, the vertical distribution of velocity u can be expressed by the Prandtl-von Kármán logarithmic law, as follows: u 1 z = ln + Bs (1) u∗ k ks p where z is the vertical coordinate; u is the time-averaged velocity in the x direction; u∗ = tb/r is the shear velocity, tb is the bed sheer stress; k is the von Kármán constant, k = 0.4; ks is the equivalent bed-roughness and is discussed in Section3; Bs = fBs (Re∗) is the roughness function, Re∗ = u∗ks/n is the roughness Reynolds number; n is the kinematic viscosity coefficient. Bs can be determined by experimental curve [1,2] or calculated approximately by the following formula [1]: h 2.55i n h 2.55io Bs = (2.5 ln Re∗ + 5.5) exp −0.0705(ln Re∗) + 8.5 1 − exp −0.0594(ln Re∗) (2) Water 2018, 10, 870; doi:10.3390/w10070870 www.mdpi.com/journal/water Water 2018, 10, x FOR PEER REVIEW 2 of 18 2.55 2.55 BRe=+−()()2.5ln 5.5 exp 0.0705 ln Re +−− 8.5{ 1 exp 0.0594 () ln Re} s ** *(2) ≥ ≡ Remarkably, when Re* 70 , the flow is the rough turbulence, and thus Bs 8.5 ; the value is independent of Re* . As a semi-empirical solution, Equation (1) is widely applied in the calculation Water 2018, 10, 870 2 of 18 of open channel flows. Figure 1. Sketch of two-dimensional (2D) open channel uniform flow.flow. The dimensionless Chézy resistance factor c is defined as cuu= [1], where u is the depth- Remarkably, when Re∗ ≥ 70, the flow is the rough turbulence, and* thus Bs ≡ 8.5; the value is −1 independentaveraged velocity. of Re∗ .Considering As a semi-empirical u being solution, equal to Equation u at the (1) isrelative widely height applied zh in the== e calculation0.368 [1], of openwhere channel h is the flows. depth, the Chézy resistance factor c is given by the following equation: The dimensionless Chézy resistance factor c is defined as c = u/u∗ [1], where u is the uh1 −1 depth-averaged velocity. ConsideringcBu==being equalln 0.368 to u at the + relative height z/h = e = 0.368 [1], κ s (3) where h is the depth, the Chézy resistanceuk* factor cis givens by the following equation: which is also the classical formula for cu. 1 h c = = ln 0.368 + Bs (3) For the two-dimensional open channelu∗ uniformk flow,ks there exists the following equation: = which is also the classical formula for c. u* ghS (4) For the two-dimensional open channel uniform flow, there exists the following equation: where S is bed slope along the channel. By combining Equations (3) with (4), u can be expressed as follows: p u∗ = ghS (4) u== cu c ghS (5) where S is bed slope along the channel. By combining* Equations (3) with (4), u can be expressed as follows:Equation (5) is in the form of the Chézy formula, in which the dimensionless resistance factor c p has a relationship with the usual Chézy coefficientu = cu∗ = givenc ghS by the following: (5) Equation (5) is in the form of the Chézy formula,Ccg= in which the dimensionless resistance factor(6) c has a relationship with the usual Chézy coefficient given by the following: The deduction of Equation (1) relies on the closure of the Reynolds stress in turbulence equations p using the Prandtl mixing length model [2] (theC details= c ong how Equation (1) is derived from the Prandtl(6) mixing length model can be seen in [3,4]). Hence, Equation (3) is also based on the Prandtl mixing lengthThe model. deduction However, of Equation with the (1) deve relieslopment on the of closure numerical of the technology, Reynolds stress the k in− turbulenceε turbulence equations model usingis more the widely Prandtl applied mixing than length the model Prandtl [2] (themixing details length on how model. Equation In recent (1) is years, derived by from employing the Prandtl the mixingk − ε turbulence length model model, can Wu be seenet al. in [5] [3 simulated,4]). Hence, the Equation flow and (3) sediment is also based transport on the in Prandtla 180° channel mixing lengthbend with model. movable However, bed with in the the laboratory; development Zhang of numerical et al. [6] technology, developed the thek two-dimensional− # turbulence model k − ε is moreturbulence widely model applied in curvilinear than the Prandtl coordinates mixing to length simulate model. the Inbehavior recent years,of turbulent by employing flow in thean openk − # ◦ turbulencechannel partially model, covered Wu et al. with [5] simulated vegetation; the Thi flow et andal. [7] sediment investigated transport the inturbulent a 180 channel flow around bend with the movablelocal scour bed occurring in the laboratory; around hydraulic Zhang et al.structures [6] developed with the the k two-dimensional− ε turbulence kmodel.− # turbulence On the basis model of inthe curvilinear three-dimensional coordinates geometry to simulate model the with behavior a geological of turbulent model, flowWang in et an al. open [8] presented channel partiallya three- covereddimensional with vegetation;k − ε turbulence Thi et al.model [7] investigated considering theBingham turbulent liquid flow character. around theBesides local these, scour occurringthe k − ε aroundturbulence hydraulic model structuresis also applied with in the solvingk − # turbulenceproblems of model. other fields, On the such basis as of air the cooling three-dimensional technologies geometry model with a geological model, Wang et al. [8] presented a three-dimensional k − # turbulence model considering Bingham liquid character. Besides these, the k − # turbulence model is also applied in solving problems of other fields, such as air cooling technologies [9], automotive applications of on-board hydrogen storage [10], internal combustion engines scale-resolving simulation [11], pollutant removal [12], etc. Water 2018, 10, x FOR PEER REVIEW 3 of 18 [9], automotive applications of on-board hydrogen storage [10], internal combustion engines scale- resolvingWater 2018, 10simulation, 870 [11], pollutant removal [12], etc. 3 of 18 In general, the treatment of each layer in the k − ε turbulence model is shown in Figure 2, in which u1 is the velocity near the bed; z1 is a selected near-bed height. This treatment assumes that In general, the treatment of each layer in the k − # turbulence model is shown in Figure2, in which u conforms to the Prandtl-von Kármán logarithmic law only within the range of z . Above this u is the velocity near the bed; z is a selected near-bed height. This treatment assumes that1 u conforms 1 1 − ε range,to the Prandtl-vonhowever, the K velocityármán logarithmic is computed law by only solving within the the turbulence range of zequation. Above closed this range, by the however, k 1 − ε model.the velocity On account is computed of that, by the solving resistance the turbulence factor c corresponding equation closed to bythe the k k − #modelmodel. is Onnot accountexactly identicalof that, theto resistancethe one calculated factor c corresponding by Equation (3). to theNotek − that# model Equation is not (3) exactly is indeed identical the tooutcomes the one correspondingcalculated by Equation to the Prandtl (3). Note mixing that length Equation model (3) is a indeedpplied to the the outcomes whole range corresponding of the depth. to the Thus, Prandtl the fundamentalmixing length reason model causing applied the to thedivergence whole range is that of there the depth. are two Thus, turbulence the fundamental models using reason different causing waysthe divergence in the closure is that of the there Reynolds are two stress turbulence in turbulence models equations.
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