Discontinuous Shallow Flow

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Presentation transcript:

Discontinuous Shallow Flow A Numerical Method for Discontinuous Shallow Flow Fritz R. Fiedler University of Idaho Department of Civil Engineering

Model Objectives Solve 2-D hydrodynamic flow equations surface flow, rainfall, infiltration stiff hyperbolic equations non-linear source term Applications rainfall-runoff process wetlands flood plains

Overland Flow

Microtopography

Equations

Numerical Challenges Non-linear hyperbolic system Strong source terms (equations stiff when h~0) Small depths / dry areas Large gradients Discontinuous flow regime

Vector Form

Vector Form

Approach Select basic numerical scheme Modify basic scheme to address problem-specific challenges Develop algorithm Develop code Test Iterate? (start simple)

MacCormack Scheme Predictor, Backward Difference Corrector, Forward Difference

Split MacCormack Scheme Lx1 Operator:

Friction Slope: stiff!

Friction Slope: Point-Implicit Treatment

Convective Acceleration Upwinding For the Lx1 operator: If flow is in the -x direction (j+1 to j)

Smoothing Function

Lateral Inflow and Infiltration

Algorithm Input Define grid Initialize Solve time loop compute lateral inflow (Newton’s Method) compute h, p, q output?

Computer Code do 102 j=1,Nx-1 delh1(j,k) = - dtohx*(pc(j,k)-pc(j-1,k))+dt*0.5*re(j,k) delp1(j,k) = - D(j,k)*dtohx* & (convacc(j,k) + g*hc(j,k)**2 - g*hc(j-1,k)**2) & - D(j,k)*dt * & ( g*(hc(j,k)+hc(j-1,k))*(z(j,k)-z(j-1,k))/hx & +Ko*0.01*pc(j,k)/8./hc(j,k)**2 & +pc(j,k)/hc(j,k)*re(j,k) ) & + D(j,k)*dt/hx**2*eps1*(pc(j-1,k)-2.*pc(j,k)+pc(j+1,k)) delq1(j,k) = - dtohx* & ( pc(j,k)*qc(j,k)/hc(j,k) & -pc(j-1,k)*qc(j-1,k)/hc(j-1,k) ) & + dt/hx**2*eps1*(qc(j-1,k)-2.*qc(j,k)+qc(j+1,k)) 102 continue

Comparative Numerical Examples Dam Break Problem Published results Physical Model Results

Dam Break Problem

Kinematic Wave Solution

Microtopographic Surface

Overland Flow Depths

Flow Depths and Velocity

Flow Channels

Overland Flow Depths

Cumulative Infiltration

Simulated vs. Experimental