Modeling Roughness Effects in Open Channel Flows D.T. Souders and C.W. Hirt Flow Science, Inc.

Transcription

1 FSI-2-TN6 Modeling Roghness Effects in Open Channel Flows D.T. Soders and C.W. Hirt Flow Science, Inc. Overview Flows along rivers, throgh pipes and irrigation channels enconter resistance that is proportional to the roghness of bonding walls. Roghness can vary considerably from smooth steel, to concrete or sand, pebbles, and even large bolders. In traditional hydralics the inflence of roghness has been cataloged in the form of a roghness coefficient based on data obtained from a wide range of field and laboratory observations. Roghness coefficients are typically defined in one of two ways: (1) Chézy s resistance coefficient, or (2) Manning s n. To nderstand these terms we mst first state the conditions nder which they are defined. For this discssion, and mch more information abot flow resistance cased by roghness, the reader is referred to Open Channel Hydralics by Ven Te Chow and pblished by McGraw-Hill, reissed Flow losses are defined in terms of a niform flow state defined as steady flow with a fixed discharge and flow depth. In practice this sally means the flow in a channel of niform cross section and having a constant slope sch that the gravitational acceleration down the slope is balanced by frictional resistance at the bondaries of the channel. Chézy derived a simple formla to describe this niform state, V = C R S, where V is the mean flow velocity in ft/s, R h is the hydralic radis in ft (i.e., flow cross sectional area divided by wetted perimeter), and S is the channel slope. C is called Chézy s coefficient of flow resistance, or simply Chézy s C. (A more general expression in which C wold be dimensionless is obtained by inserting a factor of the acceleration of gravity nder the sqare root.) The Manning formla, which is meant to do the same thing as Chézy s bt from a more empirical point of view, is / 3 1/ 2 V = Rh S, n in which n is a parameter called Manning s n. The relationship between Chézy s C and Manning s n is easily shown to be / 6 C = R h. n h 1

2 Chow gives an extensive table (see Table 5-6 and photographs of actal channels) for Manning s n that covers a wide range of artificial and natral channels. If one tries to nmerically compte flow in an open channel the qestion immediately arises of how to specify the flow loss at a rogh channel srface? Comptations of the flow profile in a channel are only affected by roghness throgh a viscos shear stress in the grid elements adjacent to the bondaries. The remainder of the flow mst adjst itself to the wall shear stresses throgh viscos forces, which for water are very small nless the flow is trblent. This means that a trblence model mst be sed in the simlations. In this Technical Note we describe a simple test problem that has been sed to stdy wall roghness effects in open channel flows. There are two isses. First, what are the proper bondary conditions to impose on the trblence transport variables at rogh bondaries? Second, how does one insre a consistency between those bondary conditions and the bondary conditions for mean qantities sch as velocity or momentm? Modeling the Effects of Roghness When a trblence transport model (e.g., k-epsilon or RNG) is sed in nmerical simlations there mst be sitable bondary conditions for the trblence qantities (i.e., trblent kinetic energy k and trblent dissipation ε) that reflect the roghness of the bondary. Typically, these bondary conditions are set sing so called wall fnctions. That is, it is assmed that a logarithmic velocity profile exists near a wall, which can be sed to compte an effective shear stress at the wall. One does this by fitting the log profile to the compted tangent velocity closest to the wall, tan, at the distance of that velocity component from the wall, y. For a smooth wall the eqation is, tan 1 ρs y = s ln + 5. κ µ smooth wall In this eqation κ is the von Karman constant and the eqation is solved for the wall shear stress velocity, s. The s vale is then sed to define wall bondary conditions for the trblent variables k and εpsilon (see, for example, Trblence Modeling for CFD, by David Wilcox and pblished by DCW Indstries, Inc., 1998). For the k-epsilon model the bondary conditions are: k = 2 s c µ, ε = κy For rogh walls the same procedre is sed, bt the logarithmic fnction has a slightly different form, 3 s 1 y tan = s ln κ rogh. rogh wall 2

3 The incorporation of rogh-wall bondary conditions into FLOW-3D has been accomplished throgh the implementation of three procedres (in code Versions beginning with 8.1): 1. Smooth and rogh wall expressions for the near-wall logarithmic velocity profiles are combined into a single expression by introdcing a modified viscosity at a bondary that reflects the enhanced momentm exchange expected becase of roghness. 2. The modified logarithmic velocity profile is sed to compte a local shear stress in elements adjacent to bondaries and this vale is then sed to define the vales of the trblent transport variables (k,e) at the bondary. (Steps 1 and 2 are in Version 8..) 3. Wall shear stresses compted from the logarithmic velocity profile are inclded in the momentm eqations for the mean flow velocity. Step 1, the blending of the smooth and rogh wall velocity profiles, is accomplished by assming that roghness makes an additional contribtion to the moleclar viscosity µ, forming an effective viscosity defined as µ eff = µ + ρ a rogh. Here s is the shear velocity tangent to the wall srface, ρ is the flid density, a is a constant, and rogh is the roghness vale inpt for the solid srface. The idea behind this proposition is that roghness elements introdce pertrbations in the flow characterized by s and rogh that increases the transfer of momentm between the flid and the bondary. The convenience of the µ eff expression can be appreciated by sbstitting it in place of µ into the smooth-wall logarithmic profile, s tan 1 ρs y = s ln + as rogh 5. κ µ + ρ combined When roghness is zero and µ eff is simply eqal to the moleclar viscosity the logarithmic velocity profile is the correct expression for a smooth wall. However, when the roghness contribtion to µ eff is mch larger than the moleclar vale the logarithmic profile redces to something that looks like the rogh wall expression, tan 1 y = s ln + 5. κ a rogh If we frther define a=.246, then this becomes the accepted expression for rogh walls, with the constant 5. replaced by 8.5. Ths, the se of an effective viscosity that is the sm of the moleclar viscosity and a roghness contribtion at a rogh-wall bondary offers a convenient way to make a continos transition between smooth and rogh wall bondary conditions. 3

4 The combined velocity profile defined above is sed by FLOW-3D to compte the shear stress at all no-slip solid bondaries in conjnction with trblence transport models. However, if the Reynolds nmber based on the shear velocity ρ s y /µ is less than 5. this indicates that the flow at the bondary is in the laminar sblayer. In this case the shear velocity is set eqal to the vale corresponding to a laminar shear stress, s = µ tan ρ y s when ρy µ 5. For laminar flow in general, or with non-transport trblence models sch as the large eddy simlation model, we se µ eff with a=.246 to compte the wall shear stress, µ eff tan τ wall = y Althogh the law-of-the-wall velocity profile is normally confined to wall regions, in channel flow this profile extends approximately throgh the entire depth of flow. In this connection, it mst be remembered that establishing this profile in the blk of the flow away from bondaries is the job of the trblence model. Unfortnately, there is no garantee that the correct logarithmic profile will be created by a given trblence model. In fact, we shall see in the next section that trblence models have definite limitations. Channel Flow Test Problem We select a generic example of an open-channel flow as a test problem to test the rogh-wall bondary condition treatment described in the previos section. The channel is assmed to be rectanglar in cross section, 5 ft wide and containing water to a depth of 2.5 ft, which gives a hydralic radis of 1.25 ft (i.e., 2.5*5/1). The slope of the channel is modeled by adjsting the components of gravity. In this case the slope was chosen to be.17 (1 degree) in or example. Since the free srface is to be flat and parallel to the bottom of the channel in niform flow conditions, the top bondary has been replaced by a free-slip rigid srface. Symmetry abot a vertical mid-plane means that only one half of the channel mst be modeled. To allow the flow to develop bt remain niform in the flow (axial) direction, periodic bondary conditions were sed in that direction. This choice greatly redces the comptational effort since it removes the axial dependency and changes a three-dimensional problem into an eqivalent two-dimensional one. A sample inpt file with data in nits of lbm, ft, s is appended to this Note. Changes to the slope, dimensions of the channel, and comptational resoltion can easily be made to this file. All comptations were carried ot ntil the average kinetic energy of the flow was no longer changing. This was typically after abot 2 to 3s of problem time. 4

5 Validation of Model Theoretical Reslts The principal reslt for making comparisons between comptations and theory (and/or experimental data) is flow rate, or eqivalently, the mean axial velocity in the channel. The Chow reference (see Overview: p.24, Eqs ) gives the following expressions for the mean velocity in smooth and rogh channels, Rhs V = s Asmooth log smooth ν Rh V = s Arogh log rogh k These expressions are arrived at by performing an approximate integration of the logarithmic velocity profile over the cross section of the channel. An exact integration cannot be done becase it depends on the details of the three-dimensional shape of the channel. This leads to the introdction of the constants A smooth and A rogh, which mst be determined from empirical data. Constant A smooth =3.25 provides a fit to the pipe flow data of Nikradse (see Chow reference for details). For rogh channels, Chow cites a stdy by Kelegan sing data by Bazin that indicates a range of vales for A rogh extending from 3.23 to Chow sggests A rogh =6.25 as a mean vale. This variation can be partially explained by the fact that the shape and arrangement of roghness elements, not jst their size, has an inflence on the flow resistance generated by the elements. Chow s tables of Manning s n reqire sers to make choices for each type of channel and srface character (e.g., concrete, wood, sand, etc.). There are many factors that can affect flow rate. Some of these identified in Chow s book inclde variations in shape and size of the channel cross section, presence of obstrctions or vegetation, and meandering. Becase of these variations, Chow s table of Manning s n lists minimm, normal, and maximm vales for each type of channel. Typical ranges for concrete channels, for instance, show variations between the minimm and maximm vales of 3% or more, while for rbble or riprap the variations are as mch as 75%. Even in clean, straight, natral streams the range is 32%. Sch variations mean that some latitde mst be allowed when comparing comptational reslts with data. If we accept Chow s average flow velocity formla for a rogh channel, and set it eqal to the definition of Manning s n, then the relationship between n and roghness k in feet is given by, Rh log k.46 R n 1 6 = h 1.88 In this expression the hydralic radis R h and k mst both be expressed in nits of feet. For se in FLOW-3D the roghness shold be converted to the length nits selected for the simlation. 5

6 Comptational Reslts for Log Profile Before doing channel flow simlations it is worthwhile to check how well the program is able to generate the logarithmic velocity profile above a solid srface. We do this with the channel-flow setp by replacing the sides of the channel with free-slip walls. The compted velocity profile depends only on the vertical coordinate z and shold reprodce the theoretical log profile, 9ys = 5.75 s log ν smooth bondary Figre 1 shows the compted velocity profile as a fnction of depth for both the k-epsilon and RNG trblence models when applied to a smooth bondary. The k-epsilon-generated profile is a reasonable fit to the theoretical logarithmic dependence. This comparison confirms that the k- epsilon trblence model is pretty mch matching one of the test cases sed to define its constant parameters. X Velocity vx. height Z for smooth channel 5 X Velocity (ft/s) Height Z (ft) Analytical FLOW-3D, k- e FLOW-3D, RNG Figre 1. Comparisons of compted and theoretical log profiles for velocity above a smooth wall. The RNG model does not do as well. It is believed that this discrepancy is partly cased by the se of approximate wall bondary conditions for the trblence variables. FLOW-3D ses the same bondary conditions for both the k-epsilon and RNG models, however, some corrections shold be made to the vale of trblence viscosity in regions of low trblence intensity. Near a rigid bondary, for example, the RNG model may need some refinement in the bondary vale of trblent viscosity. The needed correction will be inclded in Version 8.1 of FLOW-3D. A similar comparison between comptations and theory can be made for a rogh bondary. For example sing a roghness of k=.24, the compted velocity is greater than the theoretical prediction over most of the flow region. Frthermore, the compted reslts are sensitive to the grid sed for the nmerical soltions. In particlar, the velocity profiles are most sensitive to the 6

7 size of the grid element sed at the bondary, Fig.2, and least sensitive to the total nmber of grid elements, Fig.3. The best reslts are obtained with the largest grid element (.2ft) at the bondary and the total nmber of elements is not particlarly large (say 12). Velocity vs. Height Velocity vs. Height, 1st Cell Size of.1 Velocity (ft/s) Height Z, (ft) Analytical Szwall =.25 Szwall =.6 Szwall =.1 Szwall =.15 Szwall =.2 Velocity (ft/s) Height, Z (ft) Analytical 13 Cells 18 Cells 23 Cells 27 Cells Figre 2. Comparison of compted profiles with different sizes for element at the bondary. Figre 3. Comparison of compted profiles sing different resoltions, bt a fixed element size of.1ft at the bondary. These findings are well known bt not always acknowledged. Wilcox (loc.) describes the problem well and shows that accrate nmerical reslts can only be generated by ptting enogh grid elements near a bondary and then imposing the analytical soltion ( wall fnction ) at the first 4 (or more) grid points. The basic problem is the singlar behavior of the trblence variables near a wall bondary. Unfortnately, Wilcox s remedy is difficlt to implement and ndermines the seflness of a simple wall fnction bondary condition. Wilcox sggests that an alternative is to se a relatively large grid element size adjacent to a bondary ( large means covering the region of rapid variation in trblence variables and relying on the wall fnction to set the proper average vales in this region). For the channel flow test case, we se a bondary cell size of.2 with 12 cells to cover the depth of flow (2.5ft) to satisfy these conditions. Comptational Reslts for Channel A series of comptations for the complete channel were then carried ot for a range of roghness vales. A typical reslt, Fig.4, shows a weak secondary flow in the cross section of the channel that is generated by non-niform shear stresses in the region of the channel corners. 7

8 Figre 4. Cross sectional flow generated in a smooth channel. Secondary velocities are less than.3% of the mean axial flow speed. The compted average flow velocities along the channel are compared to the theoretical (empirical) reslts given by Chow in Fig.5. There is a constant offset in the compted mean velocity of abot 2ft/s for the k-epsilon model and abot 4ft/s for the RNG model. With the exception of the offset, both trblence models predict a behavior with roghness that follows the theoretical predictions very well. Mean Velocity vs. Roghness Mean Velocity (ft/s) Analytical FLOW-3D, k-e FLOW-3D, RNG Rogh, k Figre 5. Comparison of compted and theoretical average velocity in channel as a fnction of roghness. The offset in mean velocity seen in Fig.5 is associated with the constant A rogh in Chow s formla. As previosly noted, A rogh =6.25, which was sed in Fig.5 is a mean vale taken from data where A rogh varied between 3.23 an For the present test case, this variation in A rogh 8

9 translates to a variation in mean velocity from 2.65 ft/s to 14. ft/s. Based on this, we see that the compted reslts are well within the scatter of the experimental data. Comments The compted reslts are within the scatter of empirical data and, as Chow points ot, there are many physical factors in a real channel that affect its flow rate. Users are encoraged to se the new model, which simply means defining a roghness vale for the bondaries of the channel. If the vale of roghness is nknown it can be compted from Manning s n sing the formla given at the end of the Validation of Model section. Be aware, however, that compted reslts for flow rates can be no more accrate than the data on which the above formlas are based i.e., accept the vales as a decent approximation bt not the absolte trth. 9

10 APPENDIX: SAMPLE INPUT FILE FOR CHANNEL FLOW STUDY Roghness Stdy: Rogh=., k-e S=.17, Rad=1.25, K-Epsilson or RNG $xpt remark='nits are lbm,ft,s', twfin=3., delt=.1, itb=, gx=.562, gz= , ifvis=3, $limits $props rhof=1.94, $scalar $bcdata wl=4, m1=.234, wr=4, wf=1, wbk=1, wb=2, wt=1, $mesh nxcelt=2, nycelt=13, nzcelt=13, px(1)=., py(1)=., pz(1)=-.25, px(2)=2.5, py(2)=2.5, pz(2)=., $obs avrck=-2.1, nobs=1, py(3)=2.75, sizey(2)=.2, pz(3)=2.5, sizez(2)=.2, rogh(1)=., iob(1)=1, zh(1)=., iob(2)=1, yl(2)=2.5, zl(2)=., $fl flht=1., i=2., $bf $temp $motn $grafic $parts Docmentation: A free-slip top is sed to insre niform flow conditions. Periodic bondary conditions in x direction permits a steady flow in y-z to be established with minimm amont of CPU time. Symmetry is sed in y direction. 1