CIVE 530 - OPEN-CHANNEL HYDRAULICS LECTURE 11: UNSTEADY GRADUALLY VARIED FLOW

11.1  EQUATIONS OF UNSTEADY GRADUALLY VARIED FLOW

The Saint-Venant equations describe the unsteady gradually varied flow in open channels. The flow is unsteady because discharge and flow area (and stage) vary in time and space. The flow is gradually varied because the assumption of hydrostatic pressure distribution is valid (vertical accelerations are negligible). The Saint Venant equations are a set of two (mass and momentum) nonlinear (actually quasi-linear) partial differential equations describing the movement of water in one dimension (x). The complete solution of the Saint-Venant equations has not been achieved, because of their nonlinear nature. Nonlinear PDE's cannot be solved completely with currently available methods of calculus. Simplifications are necessary, such as the assumption of linearity. A linear solution is convenient, but the nonlinearity is not accounted for. In the Saint-Venant equations, the nonlinearity is usually small (quasi-linearity). The Saint-Venant equations are: Water continuity:   u (∂d/∂x) + d (∂u/∂x) + (∂d/∂t) = 0 Momentum: (1/g) (∂u/∂t) + (u/g) (∂u/∂x) + ∂d/∂x + Sf - So = 0   In the momentum equation, the term (1/g) (∂u/∂t) is the local acceleration slope. The term (u/g) (∂u/∂x) is the convective acceleration slope. The term ∂d/∂x is the slope due to the pressure-gradient. The term Sf is the friction slope. The term So is the bottom slope. The balance between these slopes (accelerations, or forces) determines the type of unsteady flow motion. The balance of friction and bottom slopes is called kinematic wave. The balance of local acceleration, convective acceleration, and pressure-gradient slopes is called inertia-pressure wave. The balance of friction, bottom, and pressure-gradient slopes is called diffusion wave. The balance of all five (two acceleration, pressure-gradient, friction, and bottom) slopes is called a dynamic wave. There are two types of acceleration because our frame of reference is fixed (Eulerian frame). If the frame of reference would be moving with the flow (Lagrangian frame), there would only be one acceleration (Newton's).

11.2  LINEAR SOLUTION OF PONCE AND SIMONS

The linear solution of the Saint-Venant equations due to Ponce and Simons makes use of the theory of linear stability, used in the study of turbulence. A particular solution is sought in sinusoidal form:   d'/do = dhat exp[ i (σhatx hat - βhatthat)]   u'/uo = uhat exp[ i (σhatx hat - βhatthat)]   in which the dimensionless wavenumber is:   σhat = (2π/L) (do/So)   and βhat is the amplification factor (related to the amount of wave diffusion).   The solution is:   i βhat2 Fo2 - i σhat2 (1 - Fo2) + 3σhat - 2βhat - 2i σhatβhatFo2 = 0   The wave celerity is: c = βhat R / σhat The dimensionless relative wave celerity is: chat  rel = (c - u) / u   The original Ponce and Simons paper.

Table 7-1 (Chow)

11.3  KINEMATIC WAVES

Kinematic waves have dimensionless relative celerity equal to 1/2 (Chezy friction in hydraulically wide channels). Kinematic waves do not attenuate, but they can undergo change of shape due to nonlinearities. Flood waves that travel with negligible attenuation (or diffusion) can be taken as kinematic waves. Kinematic waves have a single-valued rating curve (Q vs A; or Q vs y). They are derived by assuming the momentum equation to be replaced by a statement of uniform flow. The unsteady flow features (wave) are preserved through the continuity equation. Kinematic waves balance frictional and gravitational forces only. Attenuation would be caused by the pressure gradient; since there is no pressure gradient, there is no attenuation.

11.4  DIFFUSION WAVES

Diffusion waves have dimensionless relative celerity approximately equal to 1/2 (Chezy friction in hydraulically wide channels). Diffusion waves attenuate, but only a small amount. Flood waves that travel with small attenuation (or diffusion) can be taken as diffusion waves. Diffusion waves have a looped rating curve; however, the loop is small and can be neglected in practice. They are derived by assuming the momentum equation to be replaced by a statement of gradually varied flow. The unsteady flow features (wave) are preserved through the continuity equation. Diffusion waves balance frictional, gravitational, and pressure-gradient forces only. Attenuation is caused by the pressure gradient.

11.5  DYNAMIC WAVES

Dynamic waves have dimensionless relative celerity varying rapidly with wave number (wave size). Dynamic waves attenuate very quickly; some attenuate so fast that they disappear almost instantly. Flood waves that travel with large attenuation (or diffusion) can be taken as dynamic waves. This is typically the case of a flood that follows a dam breach. Dynamic waves have a looped rating curve; the loop is large and cannot be neglected. They are derived by assuming the complete momentum equation. Dynamic waves balance all forces: frictional, gravitational, pressure-gradient and inertial. Attenuation is caused by the pressure gradient and inertia.

Dam-breach flood wave propagation using dimensionless parameters.

11.6  INERTIA-PRESSURE WAVES

Inertia-pressure waves have dimensionless relative celerity equal to 1/Fo. These waves do not attenuate. These waves show up in laboratory or small-scale channels. Inertia-pressure waves balance inertial and pressure-gradient forces only.

11.7  COMPARISON BETWEEN KINEMATIC AND INERTIA/PRESSURE WAVES

These waves have the same dimensionless relative celerity when Fo= 2. Kinematic waves transport mass. Inertia-presure waves transport energy. At Fo= 2, the mass transport is being confused with the energy transport. At this point, there is surface flow instability. This signals a Vedernikov number V = 1.

11.8  ASSESSMENT OF DYNAMIC WAVES

Dynamic waves attenuate very strongly at low Froude (and Vedernikov) numbers. At V = 1, all waves do not attenuate; this leads to free surface instability. Dynamic waves are rare in nature; only in the case of a dam breach, which releases a sudden wave. There is no characteristic celerity for dynamic waves. If the dynamic wave travels with the Seddon speed, it is a diffusion wave. Most flood waves are diffusion waves.

11.9  SOLUTIONS OF UNSTEADY GRADUALLY VARIED FLOW