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2.2.1.3 DEPTH -AVERAGED SHALLOW WATER EQUATIONS The Reynolds equations that were derived in the last chapter describe the motion processes of a flow in all three dimensions. This elaborate approach is appropriate for example for the mathematical calculation of flows close to constructs or in strongly meandering rivers, so basically everywhere where the flow is dominated by three-dimensional effects. Since the numerical three-dimensional calculation is still very costly, it makes sense to reduce the Reynolds equations for calculations with simpler flow conditions. The depth-averaged two-dimensional flow equations, also called shallow water equations, provide an example. The shallow water equations are obtained, as the name suggests, by averaging the Reynolds equations over the depth. The following conditions have to be met in order for the shallow water equations to be applicable:

- the vertical momentum exchange is negligible and the vertical velocity component w is a lot smaller than the horizontal components u and v: w << u and w << v .

- the pressure gain is linear with the depth (parallel flow lines ⇒ hydrostatic pressure distribution): p(z) = γ ⋅ z with z being the depth measured from the water surface and γ = ρ ⋅ g.

These assumptions make it possible to reduce the basic system of equations to only three equations: the continuity equation (as usual) and the motion equations in direction of the x- and y-axis, as follows: x-component:

  ∂ z + h  ∂ ∂   ∂u ∂u ∂u ∂u  ( o )  ∂  ρ + u + v + w  = Fx − γ +  τ xx + τxy + τ xz  Eq. 2-170  ∂t ∂x ∂y ∂z  ∂x  ∂x ∂y ∂z 

y-component:

 ∂v ∂v ∂v ∂v  ∂ z + h  ∂ ∂ ∂v    ( o )   ρ + u + v + w  = Fy − γ +  τxy + τ yy + τ yz  Eq. 2-171  ∂t ∂x ∂y ∂z  ∂y  ∂x ∂y ∂z 

Prior to integrating the momentum equations over the depth, we will work on the continuity equation, as a small practical example to warm up. An important tool for the depth-integration are the so-called kinematical boundary conditions that give information about the change of the water surface over time.

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Kinematical boundary condition at the free surface: An expression for the surface motion is derived.

zSurface = Sohle + Wassertiefe =+ z0 h

D() z0 + h w= ,wherez()()0 + h = ft,x,y z0 + h Dt

∂+()zh∂x ∂+() zh ∂ y ∂+() zh ⇒ w =0 + 0 + 0 ,() chain rule z0 + h ∂t ∂∂ tx ∂∂ ty

∂z ∂h ∂+()zh ∂+() zh ⇒ w=0 ++ u0 + v 0 zh+ zh + zh + 0 ∂∂tt0 ∂ x0 ∂ y =0 firm river bed

∂h ∂()z + h ∂ () z + h ⇒ w= + u0 + v 0 Eq. 2-172 zh0+∂t zh 0 + ∂ x zh0 + ∂y

Kinematical boundary condition at the river bed: Ground is impermeable ⇒ no mass flux perpendicular to bed.

u⋅ nso = 0 , n so = normal vektor of the river bed

∂z0    u  ∂x    ∂z ⇒ 0  v  ⋅  = 0 Eq. 2-173   ∂y w    z0 −1     

∂z ∂ z ⇒ w= u0 + v 0 z0 z 0∂x z 0 ∂ y

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Other tools are the Leibniz theorem and the fundamental theorem of integration, that both shall be restated in this place (just to make sure):

Leibniz− theorem : ∂zho+ zh o + ∂ u ∂()z + h ∂z udz= dzu +o − u o ∫ ∫ zo + h ∂x ∂ x ∂∂ xxzo zo z o Eq. 2-174 Integration theorem : zo + h ∂u  ∫   dz= u − u ∂z  zo+ h z o zo

With these instruments we tackle the integration of the continuity equation about the vertical axis, that means between the bed z0 and the free surface z0+h (with h being the water depth):

z0 + h ∂u ∂ v ∂ w  + +  dz = 0 ∫ ∂x ∂ y ∂ z zo  

zho+∂u zh o + ∂ v zh o + ∂ w dz+ dz + dz0 = ∫∂x ∫ ∂ y ∫ ∂ z zo z o z o

∂ zo + h ∂()z + h ∂z udzu−o + u o ∫ zo + h ∂x ∂ xzo ∂ x zo first term

∂ zo + h ∂()z + h ∂z +vdzv −o + v o ∫ zo + h ∂y ∂ yzo ∂ y zo second term

Eq. 2-175 +wzo + h = 0 zo third term

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In a little different form and substituting the kinematical boundary conditions yields:

∂zho + ∂ zho + udz+ vdz ∂x∫ ∂ y ∫ zo z o

∂+()zh ∂+() zh −uo − v o + w zho +∂x zho + ∂ y zho + ∂ h = (temporal change of the surface) ∂ t

∂z ∂ z +uo + v o − w = 0 zo∂x z o ∂ y z o = 0 (no mass flux perpendiculat to the river bed)

∂zho + ∂ zho + ∂ h ⇒ udz+ vdz + = 0 ∂x∫ ∂ y ∫ ∂ t zo z o Eq. 2-176

Introduction of discharge as an integral of the flow velocity over the depth, and the depth-averaged flow velocities u and v (not to be confused with the symbols for the time-averaged fields mentioned before) we get:

zo +h zo +h q = u dz = u h und q = vdz = v h x ∫ y ∫ Eq. 2-177

zo zo

The depth-integrated continuity equation can thus finally be stated as:

∂ (u h) ∂ (v h) ∂ h + + = 0 Eq. 2-178 ∂x ∂y ∂ t

The depth-integrated continuity equation shows that the difference between the flow into and out of a volume of water comes with a change of the water depth. The derivation of Eq. 2-178 has been done without simplifying assumptions. The equation represents the conditions exactly.

Consequently we will do the depth-integration of the momentum equations. The procedure is a little lengthy, but we will go through it in detail anyway exemplary for the x-component.

∂u ∂u ∂u ∂u ∂ (z + h) 1 ∂ τ 1 ∂ τxy 1 ∂ τ 1 + u + v + w = −g o + xx + + xz + F Eq. 2-179 ∂t ∂x ∂y ∂z ∂x ρ ∂x ρ ∂y ρ ∂z ρ x

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And depth-integrated:

zo + h zo + h ∂u ∂ u ∂ u ∂ u  ∂(zo + h ) ∫+++u v wdzg  + ∫ dz = ∂∂∂∂txyz  ∂ x zo z o temporal derivation + pressure term advective term Eq. 2-180 zo + h   z0 + h 1 ∂τxx∂ τ xy ∂ τ xz 1 ∫+ +  dz+ ∫ Fdzx ρ ∂x ∂ y ∂ z  ρ zo z 0 volume forces horizontal viscouse term (1+ 2), vertical viscous term (3)

For better overview, the depth-integration will be done separately for each term. The first term includes the derivative and the advection part. Then the Leibniz theorem will be applied to both the derivative and the two horizontal advection terms, and the fundamental theorem of integration will be applied to the third advection term:

zo +h zo +h zo +h zo +h  ∂u ∂u ∂u ∂u  ∂ ∂ 2 ∂ ∫  + u + v + w dz = ∫ u dz + ∫ u dz + ∫ u vdz  ∂t ∂x ∂y ∂z  ∂t ∂x ∂y zo zo zo zo

∂ ()z + h ∂ ()z + h ∂ ()z + h − u o − u 2 o − ()u v o + ()u w Eq. 2-181 zo +h z h zo +h zo +h ∂t o + ∂x ∂y

∂ z ∂ z ∂ z + u o + u 2 o + ()u v o − ()u w zo z zo zo ∂t o ∂x ∂y

The components of this equation can be transformed so that the kinematical boundary condition can be applied:

zo+ h∂∂∂∂∂u u u u  zho+ ∂ zh o + ∂ zh o + +++u v w  dz = u dz + u2 dz + uv dz ∫∂∂∂∂∂txyzt ∫∫∫ ∂ x ∂ y zo   zo z o z o

    ∂h ∂+()zho ∂+() zh o  −+uu + v − w  zho++∂t zh o ∂ x zh o + ∂ y zho +   ∂h  = −   ∂t 

      ∂z ∂z ∂ z  +u o +uo + v o − w zo ∂t zo∂x z o ∂ y z o    =0, Eq. 2-182 becausethe =0  riverbed  do esn`t change  Lecture material – Environmental Hydraulic Simulation Page 80

The terms in the parentheses are zero because of the kinematical boundary conditions and we obtain the following for the first term:

zo +h zo +h zo +h  ∂u ∂u ∂u ∂u  ∂(u h) ∂ 2 ∂ ∫  + u + v + w dz = + ∫ u dz + ∫ u vdz Eq. 2-183  ∂t ∂x ∂y ∂z  ∂t ∂x ∂y zo zo zo

The second term, in the following referred to as pressure term, will remain unchanged.

The third term, containing the viscous terms, will partly be transformed with the Leibniz theorem and partly with the fundamental theorem of integration. First the horizontal viscous parts:

z +h z +h o τ∂ ∂ o ∂ (z + h) ∂ z xx dz = τ dz − τ o + τ o ∫ ∫ xx xx z +h xx z ∂x ∂x o ∂x o ∂x zo zo

∂ ()τ h ∂ ()z + h ∂ z = xx − τ o + τ o xx z +h xx z ∂x o ∂ x o ∂ x Eq. 2-184

zo +h zo +h τ∂ xy ∂ ∂ (z + h) ∂ z dz = τ dz − τ o + τ o ∫ ∫ xy xy z h xy z ∂y ∂y o + ∂y o ∂y zo zo

∂ ()τxy h ∂ ()z + h ∂ z = − τ o + τ o xy z h xy z ∂y o + ∂ y o ∂ y

The vertical viscous term will be integrated with help of the fundamental theorem:

z h o + ∂ τ xz dz = τ − τ Eq. 2-185 ∫ xz z +h xz z ∂ z o o zo

Depth-integrating the volume forces in direction of the x-axis with reference to Eq. 2-131 yields:

z +h z +h 1 0 1 0 F dz = ()2ρωvsin θ dz = h 2ωvsin θ Eq. 2-186 ρ ∫ x ρ ∫ z0 z0

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Considering equations Eq. 2-181 to Eq. 2-183, we get the preliminary depth-integrated momentum equation in direction of the x-axis:

∂ u h zo + h zho + zho + ∂ τ h ( ) ∂2 ∂ ∂+∂()zh 1()τ h1 ()xy +∫udz + ∫ uvdz =− g ∫ o dz +xx + ∂t ∂ x ∂ y ∂xρ ∂ x ρ ∂ y zo zo z o

τ ∂zτ xy ∂ z τ +xx o + o − xz ρz∂x ρ z ∂ y ρ z Eq. 2-187 o o o river bed

τ ∂+()zhτ xy ∂+() zh τ −xx o −o + xz ρ∂x ρ ∂ y ρ zo + h zo + h zo + h water level

+h 2ω v sin θ

The following expression will be introduced for the wind and bed shear stresses:

τ ∂+(zh) τ xy ∂+( zh ) τ τ −xx o −o +=xz wind,x ρ∂x ρ ∂ y ρρ zo + h zo + h zo + h

and Eq. 2-188

τ∂zτ ∂ z τ τ +xx o +xy o −=−xz so,x ρ∂x ρ ∂ y ρρ zo zo z o

In analogy to the time-averaging of the NS equations, a division of the momentary vertical field into a depth-integrated mean value part and a deviation of the mean value is done. For the velocity component in direction of the x-axis this division is:

u )z( = u + ~u Eq. 2-189 with u being the mean velocity over the vertical axis (not to be confused with the time-averaged mean value that has the same symbol) and ~u being the deviation of the mean velocity. The following rule of integration holds for this division:

zo+ h zho+ zh o + zh o + 2 ∫()u++= uuɶɶ() udz ∫∫ udz +uudz ɶɶ + 2 ∫ uudz ɶ zo zo z o z o = 0 Eq. 2-190

zo+ h with∫ uɶ dz= 0 ,u ɶ = u() z − u zo Lecture material – Environmental Hydraulic Simulation Page 82

Substituting the expressions for the wind and bed shear stresses as well as the parts of the division from Eq. 2-187 into Eq. 2-185 yields the following depth-averaged momentum equation in direction of the x-axis:

 2  ∂u h zo +h zo +h ∂ u h   ∂ u v h ∂ ∂ ()+ + ()+ ~u ~u dz + ~u ~v dz = ∂ t ∂x ∂y ∂x ∫ ∂y ∫ zo zo Eq. 2-191

∂ ()z o + h 1 ∂ 1 ∂ 1 −g h + ()h τxx + ()h τxy + ()τwind x, − τso x, + h2ωvsin θ ∂x ρ ∂x ρ ∂y ρ

As one of the last steps, the depth-integrated form of the continuity equation is isolated from the first three terms on the left side of the equation with help of partial differentiation.

2 ∂()uh∂ (u h ) ∂ () uvh + + ∂t ∂ x ∂ y

∂u ∂ u ∂∂ u h ∂()uh ∂ () vh  =+h hu ++ hv u  + +  Eq. 2-192 ∂t ∂∂∂∂ x ytx ∂ y   =0 depth integrated continuity equation

∂u ∂ u ∂ u  =h + u + v  ∂t ∂ x ∂ y 

Subsequently, we divide by the water depth and group the “fluctuating terms” with the viscous terms. We now get the momentum equation in direction of the x-axis:

∂ u ∂ u ∂ u + u + v = ∂ t ∂x ∂y Eq. 2-193 ∂ z + h   τ    τ   τ τ  ()o 1 ∂ xx ~ ~ 1 ∂  xy ~ ~ 1  wind x, so x,  −g + h  − u u + h  − u v +  −  + 2ωvsin θ ∂x h ∂x   ρ  h ∂y   ρ  h  ρ ρ 

For a better overview, the general form of the depth-averaged continuity equation and the depth- averaged momentum equations, the so-called shallow water equations:

∂h ∂ u i h + ( ) = 0 ∂t ∂x i Eq. 2-194

   τ τ ∂u i ∂u i ∂ 1 ∂   1 ~ ~  1 wind i, 1 so i, + u j = −g ()z o + h + h τij − u i u j + − + a F   i ∂t ∂x j ∂x i h ∂x j  ρ  h ρ h ρ   4 5  1 2 3 6 where a = „acceleration component“ of the volume forces. Fi Lecture material – Environmental Hydraulic Simulation Page 83

The individual terms will now be explained shortly. The first term on the left side is the rate of change (over time) and the second term the convective momentum transport. The right side includes the gravitation force (third term) and the fourth term is the diffuse momentum transport with:

∂u   ∂u ∂u  2  τ = µ i + ρ ν  i + j  − k δ  Eq. 2-195 (,ij m+ )t  T   ij  ∂x j   ∂x j ∂x i  3 

The fifth term is the dispersive momentum transport, which is a mathematical result similar to the Reynolds stresses – only by depth-integration. For uniform and homogeneous flows this term can be neglected. It describes the exchange processes due to vertical non-uniformities. If there is a strong secondary flow, for example because of strong meandering, the dispersive term becomes important. The sixth term combines the forces from the outside, that is bed shear stress and wind shear stress. The wind shear stress can be neglected most of the times, however.

The bed shear stress is bound to the depth-averaged velocity by a quadratic velocity law.

2 τso = cf ρ u al( lg emein ) Eq. 2-196 τso = cf ρ u u

Here, cf is the friction coefficient, which can be determined according to the flow laws of Gauckler- Manning-Strickler or Darcy-Weisbach. Darcy-Weisbach’s flow law is definitely preferred.

λ λ τ = ρ u u ⇒ c = Eq. 2-197 0 8 f 8 where drag coefficient λ can be determined with help of Colebrook & White’s formula as a function of the fields: f = cross section of form drag coefficient, Re = Reynolds number,

k s = equivalent sand roughness,

rhy = hydraulic radius.

When using the Gauckler-Manning-Strickler flow law, it is of importance that the empiric k St -value depends on Re and rhy , i.e. the discharge state. The k St -value is thus no parameter of general, global validity, but it has to be adapted to the discharge state.

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If we look at the derived shallow water equations Eq. 2-192, it is conspicuous that we have a closure problem again. In order to solve the problem, the same procedure as for the time-averaging can be applied. We simplify and summarize the fourth and fifth terms of Eq. 2-192 to:

τ − ρ~u ~u = τ = τ (,ij m+ )t i j g,ij ij τ d,ij

ν = ν + ν + ν Eq. 2-198 g m T D =0

 ∂u   ∂u i j  τij =ρν ij,g  +   ∂x j ∂x i 

We see that the total viscosity is composed of a molecular part, a turbulent part and a dispersive part.

Further transformations yield:

 ∂u   ∂u i j  τij = ρ ν ij,g  +   ∂x j ∂x i  ε = ρ ν ij ij,g Eq. 2-199

 ∂u ∂u  ⇔ τ = ε  i + j  ij ij    ∂x j ∂x i 

These simplifications make it possible to write the shallow water equations in the following form:

∂u ∂u ∂u ∂ 1 ∂ 1 ∂ + u + v = − g ()z + h + ()h τ + ()h τ ∂t ∂x ∂y ∂x o hρ ∂x xx hρ ∂y xy

1 τso x, 1 τwind x, − + + 2ωvsin θ h ρ h ρ Eq. 2-200 ∂v ∂v ∂v ∂ 1 ∂ 1 ∂ + u + v = − g ()z + h + ()h τ + ()h τ ∂t ∂x ∂y ∂y o hρ ∂x xy hρ ∂y yy

1 τso y, 1 τwind y, − + − 2ωu sin θ h ρ h ρ

And in tensor form:

∂u ∂u ∂ 1 ∂ 1 τso i, 1 τwind i, i + u i = − g ()h + z + ()h τ − + + a Eq. 2-201 j o ij Fi ∂t ∂x j ∂x i hρ ∂x j h ρ h ρ

In this equation we have mixed the dispersion part with the diffusion, so to say, by introducing a total viscosity. However, this way does not provide a pure turbulence modelling anymore, independent from the turbulence model used, since this integrated approach also models the dispersion. Lecture material – Environmental Hydraulic Simulation Page 85

In Eq. 2-198 and Eq. 2-199 all fields are time- and depth-averaged. Bars across, waves or other means of symbol declaration for the time- and depth-averaging have been left away and will not be used in the following chapters either.