The hydraulic jump in an open channel is formed when up-stream supercritical flow changes into downstream subcritical flow.Hydraulic jumps have been extensively studied owing to their fre-quent occurrence in nature and have been widely used as energydissipaters for hydraulic structures (Rajaratnam 1968; Hager 1992).The primary concern with jumps on rough beds is that the rough-ness elements located near upstream might be subjected to cavita-tion and possible erosion. In such cases, the end of the jump movesdownstream, thereby causing erosion and possibly damage to thestructure itself. The hydraulic jump over a natural rough bed ismade up of rocks consisting of various ranges of relative roughness,where the relative roughness t is the ratio of the equivalent grainroughness ks to the effective flow depth.
A sketch of a hydraulic jump over a rough bed is shown in Fig. 1,where k is the bed roughness, Fτ is the rough bed shear stressimposed on the fluid, h1 is the fluid depth at toe of the jump,h2 is fluid depth at the end of the jump, Lj is the jump length,and LR is the roller length. The application of the momentum con-servation in a control volume bounded by upstream supercriticalflow (marked 1) and downstream subcritical flow (marked 2) yieldsthe equation Π1 þM1 þ N1 ¼ Π2 þM2 þ N2 þ Fτ where Π1 andΠ2 are hydrostatic forces; M1 and M2 are the mean momentumfluxes; N1 and N2 are turbulent Reynolds normal momentumfluxes; and h1 and h2 are sequent depths, respectively, at the toe(Section 1) and exit (Section 2) of the hydraulic jump. The inte-grated bed shear stress is generally adopted as Fτ ¼ λðM1 �M2Þwhere λ is bed shear force coefficient.
For a hydraulic jump over a smooth bed rectangular channel,Belanger (1840) proposed a sequent depth ratio h2=h1 ¼ α�1 ver-sus upstream Froude number F1, which may be extended to a roughbed:
where a ¼ 8 corresponds to the channel of the smooth bed. Basedon the experimental data, Govindarao and Ramaprasad (1966)proposed a > 8 for rough bed channels. The work of Leutheusserand Kartha (1972) adopted bed shear stress as Fτ ¼ λM1 and theirprediction
e ¼ 0:0296� 0:0122αð1� αÞ ð2Þ
1Visiting Professor, Dept. of Mechanical Engineering, Korea AdvancedInstitute of Science and Technology, Daejeon 305-701 Korea, and Facultyof Engineering, Aligarh Muslim Univ., Aligarh 202002, India (correspond-ing author). E-mail: [email protected]
2Dept. of Civil Engineering, Univ. of Nebraska, Omaha, NE 68182.3Dept. of Mechanical Engineering, Korea Advanced Institute of Science
and Technology, Daejeon 305-701 Korea.Note. This manuscript was submitted on August 9, 2010; approved on
was supported by experimental data. Leutheusser and Schiller(1975) investigated the characteristics of mean turbulent motionin an artificially roughened channel in which bed roughness con-sisted of spheres (acrylic plastic balls on acrylic plastic base plates)and strips (small sheet-metal angles attached to a red wood baseplate). For a rough bed rectangular channel, Rajaratnam (1967) pre-sented interesting data for the sequent depth and jump length interms of parameter ke=h1, where ke is rough bed equivalent rough-ness and h1 is supercritical stream depth (Fig. 1). Hughes and Flack(1984) represented the relative roughness of the bed in terms of dxx(the material size of the bed mixture for which xx of material isfiner) and equivalent grain roughness height ks (assumed to be afunction of d65). Leutheusser and Schiller (1975) and Hughesand Flack (1984) analyzed their experimental data for a sequentdepth ratio that supported Eq. (2). Recently, Pagliara et al. (2008)proposed another correlation:
e ¼ �0:05kh1
½1� 1:256α�1 þ 0256α�2� þ 1:1χ�4:5;
χ ¼ σ2δσ2 � 1
ð3Þ
where k=h1 = roughness ratio; and parameter χ = function of geo-metric standard deviations δ ¼ d90=d84 and σ ¼ d84=d16 of the bedroughness. Gill (1980) predicted the effects of bed roughness uponsequent depth h2=h1 as a first perturbation over smooth surfacesequent depth α�1� ¼ ðh2=h1Þe¼0 in a rectangular channel as
h2h1
¼ α�1�
"1� 0:0082ðF2
1 � 1Þ1:5353α�2� � ð1þ 2F2
1Þ
#ð4Þ
which compares with experimental data for F1 ≤ 10. Carollo et al.(2007) considered M2 ≠ 0 and proposed sequent depth Eq. (1),where �e ¼ �λ was bed roughness effects correlated with bedroughness k=h1. Later, Carollo et al. (2009) adopted k=hc as bedroughness and proposed an empirical expression:
h2h1
� 1 ¼ffiffiffi2
pexp
�khc
�ðF1 � 1Þ0:963 ð5Þ
where hc = critical water depth, instead of h1 used earlier by Carolloet al. (2007).
The hydraulic jump on a corrugated bed has been studied byEad and Rajaratnam (2002), Tokay (2005), Yadav and Ahmad(2007), and Abbaspour et al. (2009), in which the height of corru-gation from crest to trough and wave length of corrugation play
significant roles in the corrugated beds. Mohamed-Ali (1991),Negm (2002), Izadjoo and Bajestan (2005), and Bejestan and Neisi(2009) have considered the effects of roughened-bed stilling basinon the length of a hydraulic jump in a rectangular channel.
The objective of the present work is to analyze the axial flowstructure of a turbulent hydraulic jump over a rough bed in a rec-tangular channel by using depth averaged Reynolds equations atlarge Reynolds numbers. The flow in the domain of the turbulenthydraulic jump is divided into two layers: inner and outer. In innerlayer near the wall, molecular viscosity, bed roughness, and turbu-lent process play dominant roles, which satisfies the no slip boun-dary condition over the rough bed of the rectangular channel.
Because of three-dimensional transitional roughness in stream-wise and cross streamwise directions, the roughness sublayer in theimmediate neighborhood of the channel bed would produce a com-plicated three-dimensional mean flow pattern, but slightly abovethis roughness sublayer the mean turbulent flow would be two-dimensional and dominated by the oncoming stream velocity. Theskin friction force is no greater and likely smaller than form dragowing to irregular random bed roughness. In a typical rough bed,the separation attributable to the irregular transitional bed surface isprimarily confined in the roughness sublayer. The flow separationin the roughness sublayer does not directly affect the outer layer offlow, but implicitly imposes drag force owing to skin fiction andfoam drag refereed as drag force. In fact, the foam drag that arisesowing to separation is also caused by implications of molecularkinematic viscosity effects. In the roughness sublayer, a traditionalno slip condition has be satisfied, implying small changes in veloc-ity (compared to velocity of outer stream) and consequently theFroude number based on sublayer velocity and sublayer depthin the roughness sublayer would be much less than unity. The anal-ogy of a hydraulic jump with a shock wave (Duncan et al. 1967)and analysis of the shock wave structure becomes relevant, whichfor laminar flow may be found in the work of Thompson (1972).Thus, in the outer layer, the turbulence, inertia of fluid, and im-posed drag owing to bed roughness play a major role and themolecular viscosity has a negligible effect. The matching of theouter layer to the inner layer imposes the drag owing to bed rough-ness, which has a passive role in imposing the wall shear stressowing to transitional bed roughness in the formation of the hy-draulic jump. Thus, supercritical flow F1 > 1 at the toe of the jumpchanges to subcritical Froude number F2 < 1 at the exit of thejump. In the present work, implications of the upstream Froudenumber, bed roughness drag, energy correction factor, and effective
Fig. 1. Sketch of a hydraulic jump over a rough bed: ks is the bed roughness, Fτ is the rough bed shear stress imposed on the fluid, h1 is the fluid depthat toe of the jump, h2 is fluid depth at the end of the jump, Lj is the jump length, and LR is the roller length
Reynolds normal stresses have been analyzed. The sequent depthratio h2=h1 proposed expression leads to a rational choice of param-eter e in Eq. (1). The analytical solutions of the depth averagedReynolds equation predicted velocity and jump depth profiles inthe turbulent hydraulic jump. The dynamic similarity shows thatthe roller length LR and aeration length LA are of the order of thejump length Lj, i.e., LR ¼ n1Lj and LA ¼ n2Lj, where n1 and n2, theuniversal numbers for roller and aeration lengths, are explicitlyindependent of the channel shape. The analytical expressions forjump length, roller length, and aeration length have also been pro-posed. The proposed predictions compare well with the experientialdata over rough bed rectangular channels.
Analysis of a Rough Bed Channel
The depth averaged equations of continuity and momentum for ahydraulic jump over a smooth or rough bed in a rectangular channelare (Afzal and Bushra 2002)
∂∂x
Zudz ¼ 0 ð6Þ
Z �u2 þ p
ρ� 1ρτnn � ν
∂u∂x
�dz� 1
ρ
Zτwdx ¼ C ð7Þ
where u ¼ uðx; zÞ = velocity at a point in the streamwise direction;p ¼ ρgðh� zÞ = hydrostatic pressure distribution; τnn ¼ τ xx � τ zz =effective normal Reynolds stress in the streamwise x direction;h = depth of flow in the z direction over a transitional rough bedchannel; g = acceleration attributable to gravity; τw = bed roughnessdrag force owing to friction and foam drags per unit flow depth at thebottom surface; A ¼ bh = cross-sectional area of the flow; b = widthof the channel; and C = constant of integration. In Eq. (7), the firstterm is the mean momentum flux, the second term is the hydrostaticpressure, the third term is the turbulent momentum flux, the fourthterm is the molecular viscous stress, and the fifth term is the bottombed shear stress. In terms of drag force coefficient, λ ¼ Cd=2 ¼Rτwdx=
Rρu2bdz the momentum Eq. (7) becomesZ �
ð1� λÞu2 þ pρ� 1ρτnn
�dz ¼ C ð8Þ
where p ¼ ρgðh� zÞ hydrostatic pressure distribution. The up-stream and downstream boundary conditions are x → �∞, h → h1,u → u1, τ nn → τnn1, x → þ∞, h → h2, u → u2, and τnn → τnn2,respectively. The continuity Eq. (6) and the momentum Eq. (8) havebeen integrated to yield
Uh ¼ U1h1 ð9Þ
h
�ð1� λÞð1þ βÞU2 þ 1
2gh� 1
ρTnn
�
¼ h1
�ð1� λ1Þð1þ β1ÞU2
1 þ12gh1 �
1ρTnn1
�ð10Þ
where UðxÞ = depth averaged velocity; and Tnn = depth averagedeffective Reynolds normal stress. The upstream and downstreamboundary conditions are x → �∞, h → h1, U → U1, Tnn → Tnn1,x → þ∞, h → h2, U → U2, and Tnn → Tnn2, respectively.
Jump Conditions
The continuity Eq. (9) and momentum Eq. (10) have been simpli-fied, upstream and downstream of jump, as
U1h1 ¼ U2h2 ð11Þ
h2
�ð1� λ2Þð1þ β2ÞU2
2 þ12gh2 �
1ρTnn2
�
¼ h1
�ð1� λ1Þð1þ β1ÞU2
1 þ12gh1 �
1ρTnn1
�ð12Þ
The momentum Eq. (12) may also be expressed as
h2
�fð1� λ2Þð1þ β2Þ � Γ2gU2
2 þ12gh2
�
¼ h1
�fð1� λ1Þð1þ β1Þ � Γ1gU2
1 þ12gh1
�ð13Þ
where Γ1 ¼ Tnn1ðρU21Þ�1 and Γ2 ¼ Tnn2ðρU2
2Þ�1 = effectiveReynolds normal stresses coefficients upstream and downstreamof jump, respectively.
The first invariant of the jump is the sequent depth ratio relation,obtained after elimination of velocity between Eqs. (11) and (12) as�
h2h1
�3��h2h1
�½1þ 2fð1� λ1Þð1þ β1Þ � Γ1gF2
1�
þ 2fð1� λ2Þð1þ β2Þ � Γ2gF21 ¼ 0: ð14Þ
where F1 ¼ U1=ðgh1Þ1=2 = upstream Froude number. The first twoterms in the jump Eq. (14) are expressed in factors and the remain-ing third and fourth terms from the left-hand side are moved on theright-hand side to obtain
h2h1
�h2h1
þ 1
��h2h1
� 1
�¼
�fð1� λ1Þð1þ β1Þ � Γ1g
h2h1
� fð1� λ2Þð1þ β2Þ � Γ2g�2F2
1 ð15Þ
The sequent depth Eq. (15) may be represented as
h2h1
�h2h1
þ 1
�¼ 2ð1þ eÞF2
1 ð16Þ
where function e is related to the sequent depth, drag owing to tran-sitional bed roughness, energy correction factor, and coefficient ofReynolds normal stress as
where e predicted here by Eq. (17), which differs from Eq. (2) afterLeutheusser and Kartha (1972) and Leutheusser and Schiller(1975) and Eq. (3) after Pagliara et al. (2008). The solution tosequent velocity ration from Eq. (11) yields
The second invariant of the jump is the product of the upstream anddownstream velocities, which provides
U1U2 ¼ U2c ¼ gh2c ; hc ¼
h2 þ h12ð1þ eÞ ð21Þ
where Uc ¼ ðghcÞ1=2 = critical velocity; and hc = critical depth ra-tio. Eq. (21) is analogous to the Prandtl relation for a shock wave.
The momentum Eq. (10) may be expressed as
gh2ð2F2S þ 1Þ ¼ gh21ð2F2
S1 þ 1Þ ¼ gh22ð2F2S2 þ 1Þ ð22Þ
where FS ¼ F½ð1� λÞð1þ βÞ � Γ�1=2 is defined here as aneffective Froude number, a function of conventional Froudenumber F ¼ U=
ffiffiffiffiffigh
p, bed drag force coefficient λ, kinetic energy
correction factor β and Reynolds normal stress coefficient Γ ¼TnnðρU2Þ�1. The suffixes 1 and 2 denote the upstream and down-stream values.
Jump Profile Equation
The depth averaged Reynolds Eq. (10) in the hydraulic jumpbecomes
Txx
ρ¼ ð1� λÞð1þ βÞU2 þ 1
2gh
� h1h
�ð1� λ1Þð1þ β1ÞU2
1 þ12gh1 �
Txx1
ρ
�ð23Þ
whose solution describes the turbulent flow structure in the hy-draulic jump of a rectangular rough bed channel. The hydraulicjump is analogous to a shock wave (Duncan et al. 1967; Afzaland Bushra 1999). In both situations, the upstream and downstreamflows approach certain limiting conditions: in shock waves, theseare the well known Rankine-Hugoniot relations (Thompson 1972),whereas for the hydraulic jump, these are the Belanger relations fora rectangular channel. The shock wave structure in laminar flow,reported by Thompson (1972), show that the molecular viscousnormal stress plays the dominant role. From the measurement ofResch et al. (1976) in a turbulent hydraulic jump, Afzal and Bushra(1999) proposed that the Reynolds normal stress plays a major role,as considered in Eq. (23) in terms of depth average flow. It is wellknown that the study of all turbulent flows is handicapped by theproblem of closure. Despite numerous attempts, a closure hypoth-esis that describes the essential physics in a reasonably generalfashion has yet to be constructed, and therefore analyses basedupon closure hypotheses that are not fully satisfactory are thesubject of some uncertainty. The Reynolds normal stress Tnnmay be expressed by a simple closure eddy viscosity model interms of gradient of depth averaged axial velocity with respectto axial distance as
Tnn ¼ ρντ∂U∂x ð24Þ
where ντ = kinematic eddy viscosity of flow. In the present work,the jump is associated with the outer layer, where the bed roughnessboundary condition imposed by the inner layer has a passive role inthe formation of the hydraulic jump. In the outer layer, the eddyviscosity ντ from the dimensional argument is proportional to avelocity scale △U and length scale △h, and a number of propor-tionality. In a traditional work, Clauser (1956), in the outer layer ofa turbulent boundary layer over a smooth surface, adopted thevelocity scale △U ¼ Uc and △h ¼ δ�, where Uc is the velocity
at the edge of turbulent boundary layer and δ� is the displacementthickness of turbulent boundary layer displacement thickness, thusντ ¼ ϵUcδ�. From extensive experimental data, Clauser proposedϵ ¼ 0:18, a universal number, and Townsend (1976) proposedϵ ¼ 0:2. The dimensional eddy viscosity ντ, in a hydraulic jumpformed in the streamwise x-direction, was proposed by Afzaland Bushra (2002) on a smooth channel bed in a proper analogywith Clauser (1956) for a turbulent boundary layer formed in thenormal y-direction. In a hydraulic jump, kinematic eddy viscosityντ depends on drag owing to transitional bed roughness, kineticenergy correction factor, overall velocity jump △U ¼ U1 � U2,jump depth △h ¼ h2 � h1, and a universal number ϵ independentof channel geometry and bed roughness. Thus kinematic eddy vis-cosity ντ may be expressed as
If the function ð1� λÞð1þ βÞ � Γ remains invariant at two ends ofthe jump, then Eq. (17) is simplified and the sequent depth ratioEq. (16) may be expressed as
In the present work, the variation of bed drag coefficient λ, effectiveReynolds normal stress coefficient Γ, and kinetic energy correctionfactor β across the jump have also been neglected, i.e., λ≈λ1 ≈ λ2, β ≈ β1 ≈ β2, and Γ≈ Γ1 ≈ Γ2, first owing to the in-variance of e at two ends (upstream and downstream of the jump),and second, because an additional condition across the jump wasadopted for analytical integration of the hydraulic jump equationacross the jump. In terms of effective upstream Froude numberFS1 ¼ ½ð1� λÞð1þ βÞ � Γ�1=2F1, the sequent depth ratio Eq. (28)becomes
which is explicitly independent of bed roughness, analogous to theBelanger (1840) relation. The second invariant is the product ofupstream and downstream velocities across the hydraulic jump:
U1U2 ¼ U2c ¼ ghc; hc ¼
hmð1� λÞð1þ βÞ � Γ
;
hm ¼ 12ðh1 þ h2Þ ð30Þ
The turbulence level attributable to Reynolds normal stresses Γmay be neglected, and the sequent depth Eq. (28) becomes
and the effective upstream Froude number becomes FS1 ¼½ð1� λÞð1þ βÞ�1=2F1. In this situation, the governing Eq. (26)of the hydraulic jump in the axial direction velocity and depthprofiles becomes
ϵðU1 � U2Þðh2 � h1Þh∂U∂x ¼ hU2 � h1U2
1 þg2
h2 � h21ð1� λÞð1þ βÞ
ð32Þ
where Uh ¼ U1h1. The flow at the toe of the jump is supercritical(upstream Froude number F1 > 1), and at the exit of jump is sub-critical (downstream Froude number F2 < 1). The length of ahydraulic jump is often an important factor to know when consid-ering the design of structures like settling basins. The length of ahydraulic jump is often hard to measure in the field and duringlaboratory investigations owing to the sudden changes in surfaceturbulence level and because of the formation of roller eddies.The jump length may be defined as the distance measured fromthe front face of the jump to the point on the surface immediatelydownstream of the rollers. The most common definition is by pass-ing a tangent to the depth (of velocity) at the center and finding outwhere the theoretical upstream and downstream depth conditionsare met. Thus, hydraulic jump length Lj ¼ ðh2 � h1Þ=ð∂h=∂xÞmis shown in Fig. 2(a), and is defined as the ratio of the jumpin fluid depth h2 � h1 to the slope of the jump depth profileð∂h=∂xÞm at mean depth profile h ¼ hm ¼ ðh2 þ h1Þ=2. Likewise,the jump thickness based on the fluid velocity profile Lu ¼ ðU1 �U2Þ=ð∂U=∂xÞm shown in Fig. 2(b) if the ratio of the velocity jumpU1 � U2 to the slope of the jump velocity profile ð∂U=∂xÞm atmean velocity U ¼ Um ¼ ðU1 þ U2Þ=2.
Velocity Distribution in the Jump
The jump Eq. (32) is simplified in terms of the nondimensionalaxial velocity profile V ¼ ðU � U1Þ=ðU2 � U1Þ and the nondi-mensional streamwise coordinate X ¼ x=h2 to provide
ϵ½1þ Vðα� 1ÞV � ∂V∂X ¼ Vð1� VÞ�1þ 2α1� α2 � V
�ð33Þ
subjected to the upstream and downstream boundary conditionsX → �∞, V → 0, X → ∞, and V → 1, respectively. The closedform solution yields
V�ð2þαÞð1� VÞ1þ2α�1þ 2α1� α2 � V
�1�α
¼ exp
��ð1þ 2αÞð2þ αÞ
1� α2
X þ Lvϵ
� ð34Þ
The constant Lv in the jump profile Eq. (34) is related to the originof the hydraulic jump, and is left underdetermined, which may beestimated from experimental data. The length of the hydraulicjump in terms of velocity U is shown in Fig. 2(b). In terms ofnondimensional V , the jump length may be expressed as Lu ¼ðV2 � V1Þ=ð∂V=∂xÞm, where ð∂V=∂xÞm is the slope of surface pro-file at mean velocity Vm ¼ ðV2 þ V1Þ=2. The length of jump Lufrom Eq. (33) becomes
Luh2
¼ 4ϵð1� αÞ ð1þ αÞ21þ 4αþ α2 ð35Þ
Depth Distribution in the Jump
Based on the nondimensional depth profile η ¼ ðh� h1Þ=ðh2 � h1Þ, the jump Eq. (32) may be expressed as
ϵð1þ αÞ ∂η∂X ¼ ηð1� ηÞ�η þ 1þ 2α
1� α
�ð36Þ
subjected to upstream and downstream boundary conditionsX → �∞, η → 0, X → þ∞, and η → 1, respectively. The solutionof the jump profile Eq. (36) becomes
η�ð2þαÞð1� ηÞ1þ2α�η þ 1þ 2α
1� α
�1�α
¼ exp
��ð1þ 2αÞð2þ αÞ
1� α2
X þ Lηϵ
� ð37Þ
For a large effective upstream Froude number FS1, the pa-rameter α → 0 and the asymptotic Eq. (37) yield X þ Lη ¼0:5 ln½η2=ð1� η2Þ�, which is independent of α or FS1 (as in aclassical hydraulic jump on the smooth bed of a rectangular chan-nel). The constant Lη in the solution is related to the origin of thehydraulic jump and left underdetermined in the analysis, but maybe estimated from experimental data of the initial condition ofthe jump.
Fig. 2. The turbulent hydraulic jump length over a rough bed rectan-gular channel: (a) Lj ¼ ðh2 � h1Þ=ðdh=dxÞm, ratio of jump in axialdepth h2 � h1 with slope ðdh=dxÞm at mean depth of fluid hm ¼ðh2 þ h1Þ=2; (b) Lu ¼ ðU1 � U2Þ=ðdU=dxÞm, the ratio of axial ve-locity jump U1 � U2 to slope ðdU=dxÞm at mean velocity Um ¼ðU2 þ U1Þ=2
where FS1 ¼ ½ð1� λÞð1þ βÞ�1=2F1 = effective upstream Froudenumber.
The roller length, LR, is the horizontal distance between the toesection with the flow depth h1 and the roller end. The analysis ofthe depth averaged Reynolds equations of mean turbulent flow in achannel is sufficiently general. From the dynamic similarity, it ispostulated that the length scale of the roller LR is of the orderof the jump length scale Lj, i.e., LR ¼ n1Lj and ϵR ¼ n1ϵ, wheren1 is a universal number explicitly independent of channel shape.The roller length in view of this postulate yields
The aeration length LA is defined as the reach between the upstreamend of the longer wing and the location at which air clouds have leftthe flow, according to the analysis of depth averaged Reynoldsequations of mean turbulent flow in a channel of arbitrary crosssection. From the dynamic similarity, it is also postulated thataeration length LA is of the order of the length scale of the jumpLj, i.e., LA ¼ m2Lj and ϵA ¼ m2ϵ where m2 for roller length is auniversal number explicitly independent of channel shape. The ex-pression for aeration length becomes
Regarding the experimental data in a rectangular smooth channelfor 2 < F1 < 15, Afzal and Bushra (2002) proposed Λ ¼ 8=3,ϵΛ ¼ 6:9, and ϵ ¼ 2:58 for jump length, ϵRΛ ¼ 5:2 and ϵR ¼ 1:95for roller length, and ΛϵA ¼ 3:90 and ϵA ¼ 2:58=0:66 for aerationlength. The universal constants ϵ, ϵR and ϵA are related as
ϵ ¼ 1:32ϵR ¼ 0:66ϵA ¼ 2:58 ¼ 6:90Λ
ð44Þ
which are independent of the shape of the channel cross section(Bushra and Afzal 2006). The sequent depth ratio Eq. (31), jump
length Eq. (38), roller length Eq. (40), and aeration length Eq. (42)may also be expressed as
h2h1
� 1 ¼ ϕ;Ljh1
¼ 83ϵϕ;
LRh1
¼ 83ϵRϕ;
LAh1
¼ 83ϵAϕ
ð45Þ
The function ϕ sequent depth Eq. (31) is expanded in the powers ofFS1 � 1 by a Taylor series as
ϕ ¼ 43ðFS1 � 1Þ þ 2
27ðFS1 � 1Þ2 � 32
739ðFS1 � 1Þ3 þ � � � ð46Þ
Based on the leading order term in Eq. (46), the sequent depth ratiobecomes
h2h1
� 1 ¼ 43ðFS1 � 1Þ ð47Þ
and jump length, roller length, and aeration length become
Ljh1
¼ ϵ329ðFS1 � 1Þ ð48Þ
LRh1
¼ 329ϵRðFS1 � 1Þ ð49Þ
LAh1
¼ 329ϵAðFS1 � 1Þ ð50Þ
Eq. (47) may be compared with empirical Eq. (5) by Carollo et al.(2009) for sequent depth ratio. The roller length LR empiricalrelations proposed by Carollo et al. (2007) are
LRh1
¼ 6:525
�exp
��0:60
kh1
��ðF1 � 1Þ;
LRh1
¼ 2:244
�h2h1
��1:272ð51Þ
The work of Pagliara et al. (2008) proposed the sequent depthrelation Eq. (1), subjected to Eq. (3), and the empirical relationsfor jump length Lj and roller length LR are
Ljh1
¼ 5;000ð1� Fm1 Þ; m ¼ �0:0086 exp
�� 0:466
χ
�ð52Þ
LRh1
¼�27:457
kh1
� 73:517þ 20χ2:1
��1F1
� 12
�ð53Þ
The empirical Eq. (5) by Carollo et al. (2009), Eq. (51) by Carolloet al. (2007), and Eqs. (52) and (53) by Pagliara et al. (2008) arevalid for small values on F1 � 1. On other hand, the writers’ firstorder analytical predictions in Eqs. (47)–(50) in terms of parameterFS1 � 1 are explicitly independent of the bed roughness of thechannel.
Results and Discussion
The analysis of the hydraulic jump over a rough bed rectangularchannel has been presented. The solution Eq. (31) obtained forsequent depth ratio is described as follows:
and has been compared with the experimental data of the hydraulicjump. In terms of FS1 ¼ ½ð1� λÞð1þ βÞ�1=2F1, the effective up-stream Froude number, the sequent depth ratio becomes
The relative roughness is connected to bed drag force coefficient λthrough sequent depth ratio for each Froude number. Thus, bedroughness drag coefficient λ is estimated from sequent depthEq. (54) for each set (h2=h1, F1) of experimental data for prescribedroughness, after neglecting the β effect. The effects of relativeroughness drag are considered in terms of the roughness parameter
k=h1 and alternate parameter k=hc, where hc ¼ ðq2=gÞ1=3 ¼ h1F2=31
is the critical depth. The data of bed roughness drag coefficient λ isshown in Fig. 3(a) against relative roughness k=h1. Therefore, thecurrent work proposes the following prediction for bed drag forcecoefficient:
λ ¼ 1� exp
��0:55
�kh1
�0:75
�ð56Þ
which is also shown Fig. 3(a), and compares well with the exper-imental data. Furthermore, a simple linear relation
λ ¼ 0:45kh1
ð57Þ
which is also shown in Fig. 3(a) compares slightly better with theexperimental data for k=h1 ≤ 1. The relation proposed by Carolloet al. (2007)
λ ¼ 2πarctan
�0:8
�kh1
�0:75
�ð58Þ
is also shown in Fig. 3(a), and compares well with Eq. (56). Thebed roughness drag coefficient λ against alternate roughnessparameter k=hc is shown in Fig. 2(b) along with the proposedrelations
λ ¼ 1� exp
��2:05
�khc
�0:75
�ð59Þ
λ ¼ 2πarctan
�1:4
�khc
�0:75
�ð60Þ
that also compare well with the experimental data. Furthermore,a simple linear prediction
λ ¼ 1:5khc
ð61Þ
which is also shown in Fig. 3(b) agrees with the data for k=hc ≤ 0:3.The experimental data shown in Figs. 3(a) and 3(b) have appreci-able scatter, but these predictions nearly represent the mean of thedata. The roughness drag coefficient λ in terms of roughness scalek=h1 from the experimental data in Fig. 3(a) show that Eqs. (56)and (57) describe the data roughly o of the same order. Further-more, the roughness drag coefficient λ in terms of the roughnessscale k=hc, from experimental data in Fig. 3(b) shows that theEqs. (59) and (60) also describe the data, roughly to the same order.In the present work, the bed roughness drag Eq. (55) is adopted forcomparison of the writers’ prediction with the experimental dataon jump characteristics over transitional rough beds. The sequent
Fig. 3. The prediction of bed roughness drag coefficient λ versus re-lative roughness from experimental data for transitional bed roughnessin a rectangular channel: (a) k=h1; (b) k=hc
Fig. 4. Comparison of experimental data with sequent depth ratioh2=h1 versus upstream Froude number F1 Eq. (51) with roughness dragcoefficient Eq. (52) attributable to transitional bed roughness in a rec-tangular channel
depth ratio h2=h1 with the upstream Froude number F1 shown inFig. 4 from the experimental data for relative bed roughness0 ≤ k=h1 < 2:5 and bed drag coefficient 0 ≤ λ < 0:7 has beencompared with Eq. (54) on the sequent depth ratio involving bedroughness drag coefficient from Eq. (56). The comparison of thisstudy’s prediction for the universal sequent depth ratio h2=h1Eq. (55) based on the upstream effective Froude number FS1, shownin Fig. 5, compares with the experimental data for all types of bedroughness.
In a rectangular channel, the nondimensional jump length Lj=h2and Lj=h1 versus upstream Froude number F1 from experimentaldata are shown in Figs. 6(a) and 6(b), which reveals the dependenceon drag owing to transitional bed roughness. For large values ofupstream Froude numbers F1 → ∞, the jump length Lj=h2 ap-proaches a constant value that depends on bed roughness. Thisstudy’s prediction of jump length from Eq. (38) with 8ϵ=3 ¼ 6:9is also shown in the same figure for relative roughness k=h1 ¼0, 1=4, 1=2, 3=4, 1, 2, 2.5, and 3. Accurate data is needed, particu-larly in terms of certain fixed value of k=h1, rather than particular
Fig. 5. Comparison of experimental data with this study’s universalprediction of sequent depth ratio h2=h1 versus effective upstreamFroude number FS1, Eq. (28), for all bed roughness in a rectangularchannel
Fig. 6. Comparison of Eq. (39) for jump length versus upstreamFroude number F1 with transitional rough bed experimental data ina rectangular channel: (a) Lj=h2; (b) Lj=h1
Fig. 7. (a) Comparison of universal Eq. (37) for jump length Lj=h1versus sequent depth ratio h2=h1 with experimental data for all bedroughness in a rectangular channel; (b) comparison of this study’s uni-versal predictions with experimental data for jump length Lj=h1 versus
effective upstream Froude number FS1½¼ F1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� λÞð1þ βÞ � Γp � for
values of k. The nondimensional jump length Eq. (39) in terms ofeffective Froude number FS1 is a universal relation that explicitlydoes not depend on the bed roughness. The jump length Lj=h1 ver-sus h2=h1 and FS1 from Eq. (39) is also a universal relation. To testthis universal proposition, the same experimental data are alsoshown in Fig. 7(a), where the data scatter fits well with coeffi-cient 8ϵ=3 ¼ 6:3. Subramanya (1998) suggested the jump lengthLj=h2 ¼ 6:1 over a smooth bed rectangular channel, which practi-cally remains constant for F1 > 5, whereas Elevatorski (1959) pro-posed Lj=h2 ¼ 6:9. Clearly, better experiments are needed withrespect to bed roughness for particular fixed values of relativeroughness k=h1 for moderate and higher Froude numbers. The roleof the effective upstream Froude number FS1 based on bed rough-ness is investigated for jump length Lj=h1. The experimental data ofHughes and Flack (1984) and Ead and Rajaratnam (2002) for jumplength Lj=h1 versus FS1 are shown in Fig. 7(b); which also providesstrong support for the writers’ universal relations, explicitly inde-pendent of bed roughness. The leading term approximations in(FS1 → 1) in Eq. (43) for jump length Lj=h1 is also shown inFig. 7(b) and is in good agreement with the experimental data.
In rectangular channels the nondimensional roller length LR=h2and LR=h1 against the upstream Froude number F1 are shown inFigs. 8(a) and 8(b), respectively, from the rough bed data of Hughesand Flack (1984), Ead and Rajaratnam (2002), and Carollo et al.
(2007). For large values of upstream Froude numbers F1 → ∞, theroller length LR=h2 approaches a constant universal value thatdepends on the bed roughness. This study’s prediction of rollerlength from Eq. (41) is also shown for k=h1 ¼ 0, 1=4, 1=2, 3=4,1, 2, 2.5, and 3. Accurate data is needed, particularly in termsof a certain fixed value of k=h1, rather than particular values of k.The nondimensional roller length Eq. (40) is a universal relationthat does not depend on bed roughness. The roller length LR=h1versus h2=h1 from Eq. (36) is also a universal relation. To test thislinear proposition, the same experimental data is shown in Fig. 9(a);within the scatter, the data fit well with coefficient 8ϵR=3 ¼ 4:2.The role of the effective upstream Froude number FS1 on rollerlength LR=h1 from the experimental data of Carollo et al. (2007)and Ead and Rajaratnam (2002) is shown in Fig. 9(b), which alsoprovides strong support for this study’s universal relations, explic-itly independent of bed roughness. The leading term perturbationsolution in parameter FS1 → 1 from roller length Eq. (44) in termsof LR=h1 is also shown in Fig. 9(b) and is in good agreement withthe experimental data.
Fig. 8. Comparison of Eq. (41) versus upstream Froude number F1
with a transitional rough bed experimental data in rectangular channelfor two jump lengths: (a) LR=h2; (b) LR=h1
Fig. 9. Comparison of this study’s universal predictions versus effec-tive upstream Froude number FS1½¼ F1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� λÞð1þ βÞ � Γp � with
experimental data for various bed roughness of a rectangular channel:(a) jump length LR=h1 versus sequent depth ratio h2=h1; (b) rollerlength LR=h1
1. The turbulent hydraulic jump theory over a rough bed rectan-gular channel has been proposed from depth averaged analysisof the Reynolds momentum equation. The bed shear stress at-tributable to the transitionally rough bed surface of the channelis considered while integrating the depth averaged Reynoldsequations. The flow at the toe of the jump is supercritical (up-stream Froude number F1 > 1), which at the exit of jump issubcritical (downstream Froude number F2 < 1).
2. The skin friction force is not greater and likely smaller thanform drag owing to irregular random bed roughness. In a ty-pical bed roughness, the separation attributable to irregulartransitional bed surface is primarily confined in the roughnesssublayer. The flow separation in the roughness sublayer doesnot directly affect the flow in the outer layer, but implicitlyimposes drag force owing to the bed friction and foam dragowing to bed roughness. In fact, foam drag that arises becauseof separation flow on a transitional bed is also caused by thefluid molecular kinematic viscosity effects.
3. The flow invariant relations in the jump are attributable to theupstream and downstream fluxes where sequent depth Eq. (18)and velocity jump Eq. (19) are based on parameter e defined byEq. (17). For a particular case if ð1� λÞð1þ βÞ � Γ is invar-iant at two ends of the jump, then sequent depth Eq. (25) andvelocity jump ratios are very much simplified. The sequentdepth ratio and critical depth depend on bed roughness dragcoefficient λ, in addition to the upstream Froude number F1for a particular channel shape. In terms of effective Froudenumber FS ¼ ½ð1� λÞð1þ βÞ�1=2F, the sequent depth ratio,and other hydraulic jump characteristics over the rough bedcan be deduced from the classical hydraulic jump over smoothbeds, provided that the upstream Froude number F1 is replacedby the upstream friction Froude number FS1. The Belanger’sjump condition in turbulent flow is extended for transitionalbed roughness, kinetic energy correction factor, and turbulentnormal stress fluctuations of the momentum transfer.
4. The bed roughness drag coefficient λ as a function of bedroughness scale k=h1 has been predicted by Eqs. (55) and(57), which describe the data to the same level of accuracy.Furthermore, λ as a function of alternate bed roughness scalek=hc is predicted by Eqs. (55) and (56), which also describe thedata to the same order. In the present work, Eq. (55) is adoptedfor the prediction of jump characteristics.
5. The depth averaged Eq. (23) over a rough channel bed is closedby a simple eddy viscosity model Tnn ¼ ρντ∂U=∂x. The eddyviscosity expression ντ ¼ ϵð1� λÞð1þ βÞðU1�U2Þðh2�h1Þincorporates the effects of transitional roughness attributableto bed roughness drag coefficient λ, kinetic energy correctionfactor β, overall jump velocity scale △U ¼ U1 � U2, andjump length scale △h ¼ h2 � h1. Here ϵ is a universal con-stant, independent of channel geometry and bed roughness.
6. The length of a hydraulic jump is often hard to measure in thefield and during laboratory investigations because of the sud-den changes in surface turbulence, in addition to the formationof rollers and eddies. The most common definition is by pas-sing a tangent to the depth (of velocity) at the center and find-ing out where the theoretical upstream and downstream depthconditions are met. The jump thickness based on axial velocityprofile Lu ¼ ðU1 � U2Þ=ð∂U=∂xÞm is the ratio of the jump invelocity U1 � U2 to the velocity gradient ð∂U=∂xÞm at meanvelocity U ¼ Um ¼ ðU1 þ U2Þ=2 in the jump. Likewise, thehydraulic jump length Lj ¼ ðh2 � h1Þ=ð∂h=∂xÞm is the ratioof the rise of fluid depth h2 � h1 to depth gradient ð∂h=∂xÞm
at mean depth h ¼ hm ¼ ðh2 þ h1Þ=2. The theory predictsLj ¼ ϵΛð1� αÞ, where Λ ¼ 8=3 and bed roughness data agreewith ϵ ¼ 2:58, the smooth bed value (Afzal and Bushra 2002).The data show that the validity of the dynamic similarity thatthe roller length LR and aeration length LA are of the orderof the jump length Lj, and that the constant of proportionalityis explicitly independent of channel shape and bed roughness.The roller length, LR ¼ ϵRΛð1� αÞ, and aeration length LA ¼ϵAΛð1� αÞ are of the same order asLj, which leads to eddy visc-osity universal number ϵ¼ 1:32ϵR ¼ 0:66ϵA ¼ 2:58¼ 6:9=Λ,the same as the smooth bed channel.
7. The solution of the jump profile η versus X for a rough bedrectangular channel has been proposed. The jump length Ljrelation is obtained in analogy with the shock wave thickness,leading to Lj=h2 ¼ ϵΛð1� αÞ, which is explicitly independentof λ the drag of bed roughness, but depends on channel geo-metric shape factor, which for a rectangular channel isΛ ¼ 8=3. The jump length Lj=h2 versus α and Lj=h2 versus FS1are universal relationships that are explicitly independent of λthe drag attributable to bed roughness and β the energy correc-tion parameter. However, the jump length Lj=h2 versus F1 is anonuniversal relation that depends on drag owing to bed rough-ness and energy correction factors in the channel. The expres-sions for the roller length LR=h2 and the aeration length LA=h2turn out to be analogous with the jump length Lj=h2, which interms of F1 depend on bed roughness but are universal in termsof FS1 as explicitly independent of bed roughness and energycorrection factor.
8. The proposed theory for rough bed rectangular channels pre-dicted universal solution depth profile, sequent depth ratio,jump length, and roller lengths in terms of effective upstreamFroude number FS1 and are explicitly independent of bedroughness. The experimental data of Hughes and Flack (1984),Carollo et al. (2007, 2009), and Ead and Rajaratnam (2002)support the proposed universal theory. Thus, the results forrough bed channels can be directly deduced from the classicalsmooth bed hydraulic jump theory, provided the upstreamFroude number F1 may be replaced by the effective upstreamFroude number FS1.
Notation
The following symbols are used in this paper:AðxÞ = bhðxÞ area of flow in rectangular channel;
b = width of the rectangular channel;Cd ¼ 2
Rτwdx=
Rρu2bdz = coefficient of drag attributable to
channel bed;e = Eq. (17) in sequent depth Eq. (16) of the hydraulic jump;
e ¼ ð1� λÞð1þ βÞ � Γ� 1 = assumed invariant function acrossthe jump, which includes special case λ≈ λ1 ≈ λ2,β ≈ β1 ≈ β2, and Γ≈ Γ1 ≈ Γ2;
LR = roller length in the formation of the hydraulic jump;LV = velocity length attributable to velocity profile in the jump;
p ¼ ρgðh� zÞ = hydrostatic pressure distribution;q ¼ Q=b = discharge per unit width of the channel flow;Tnn ¼ Txx � Tzz = effective depth averaged Reynolds normal stress
in the hydraulic jump;Tnn ¼ ρντ∂U=∂x = eddy viscosity closure model in the hydraulic
jump;Txx ¼ ð1=hÞ R τ xxdz = depth averaged Reynolds normal stress
τ xx ¼ �ρu0u0 in the streamwise x-direction;Tzz ¼ ð1=hÞ R τ zzdz = depth averaged Reynolds normal stress
τ zz ¼ �ρw0w0 in the z-direction;UðxÞ ¼ ð1=hÞ R udh = cross-sectional averaged velocity in the
x-direction;UhðxÞ = free surface velocity at z ¼ hðxÞ;uðx; zÞ = local velocity at a point in the streamwise
the jump;Λ = constant in the jump based on shape of the channel;
λ ¼ Cd=2 = drag coefficient of the channel in the hydraulic jump;ν = molecular kinematic viscosity of fluid;ντ = eddy viscosity of flow in the jump;ρ = fluid density;
τw = bed roughness drag force per unit flow depth at thebottom;
τ xx,τ zz
= Reynolds normal stresses in streamwise and normaldirections; and
τ xz = Reynolds shear stress in x-z plane.
Subscripts
1 = upstream of jump; and2 = downstream of jump.
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