12.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 I. Flow Mechanics 1. Conservation of Momentum Objective: derive basic open channel flow equations. Force balance for fluids. Derivation of Boundary Shear Stress (τb) and factors that control its magnitude in natural flows. Definitions: Linear Momentum = mass -velocity product: Change in momentum = acceleration: ddtmu( )= mdudt= mamu 1A. Kinematics: Isaac Newton (1687) Newton’s Second Law: “change in linear momentum is equal to the sum of forces acting on the body” F!= ma1B. Conservation of Momentum for Fluids Chauchy’s First Law: generalization of Newton’s Second Law for a parcel of fluid -- momentum balance for a unit volume of fluid (Fv = force per unit volume) Consider a volume of fluid (flow depth h and unit bed area (Δx, Δy) of density ρ moving with mean velocity u : Fv=!du dt"We will derive the sum of forces acting on a 1-dimensional flow (velocity only varies in the downslope or x-direction) for simplicity. Variables: flow depth (h), bed surface slope (α)= angle between z and g, and mean velocity ( u ), where mean velocity implies the depth-averaged value. 112.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Note that as we will see, water responds to the surface slope, which in the simplest case of uniform depth is equal to the bed slope (α). As bed slope is far easier to measure, it is often used as an approximation of the water surface slope. 212.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Forces acting on a volume of fluid include: Gravity, Pressure Gradient, Bed Friction 1B.1 Gravity (body force) Weight = ρgΔxΔyh (acts in the vertical direction) Driving force = downslope component of weight = ρgΔxΔyh sinα (Note: Normal force = ρgΔxΔyh cosα) Driving force per unit volume = ρgΔxΔyh sinα /(ΔxΔyh) = ρgsinα 1 Β . 2 Pressure gradient (pressure (p) = force per unit area) Hydrostatic pressure = weight of the overlying water column per bed area = ρgh. Here we make the small angle approximation that cosα ~ 1, so the fact that h is not measured in the vertical is negligible. This Hydrostatic pressure acts on both the upstream and downstream sides of the unit volume = force per area (Δyh). There is a net force on the volume only in the presence of a pressure gradient. The pressure gradient is given by the difference between pressure felt on the upstream and the downstream edges of the volume, divided by the width (Δx) of the volume, thus giving a net force per unit volume: p x( )! p x + "x( )"x=#gh x( )!#gh x + "x( )"x=#g"h"x=#g$h$x1B.3 Bed Friction Bed friction is described by the Shear Stress (τb) acting on the bed. The fluid exerts a shear stress on the bed (oriented downstream), and the bed exerts this same shear stress on the fluid (oriented upstream). Bed friction is the primary source of resistance to flow. Note that a stress is defined as a force per unit area. Thus, bed friction force per unit volume of fluid (fv) is given by the basal (or boundary) shear stress divided by flow depth (h): fv=!bh3!bh="gsin#du =!u !tdt +!u !xdx12.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 The condition of an unaccelerated fluid (steady, uniform velocity) of uniform flow depth requires that bed friction must balance the gravitational driving stress: which is identical for the force balance on a resting block on an inclined plane familiar from freshman physics. A more formal proof of this follows below. 1B.4 Momentum Equation for Fluids (1-dimensional flow) Chauchy’s First Law: Rate of change of momentum balances sum of forces !du dt= Fv"= gravitational driving force +/-pressure gradient -bed friction Signs: gravity always drives flow (positive), pressure gradient can either drive flow (depth decreases downstream) or resist flow (depth increases downstream), and bed friction always resists flow (negative). From 1B.1, 1B.2, 1B.3: !du dt=!gsin"#!g$h$x#%bhNote that the sign on the pressure gradient term creates the desired effect. 1C. Temporal and Spatial Momentum Changes The term !du dtdenotes the TOTAL or MATERIAL derivative of momentum, encapsulating BOTH temporal and spatial changes. The total derivative is also called the material derivative because it tracks changes in momentum in a frame of reference that follows a given parcel of water, thus both temporal and spatial variations in flow velocity, and thus momentum are “felt”. How can you isolate temporal and spatial u = u x, t( )changes? hint: the mean flow velocity ( ) MUST matter, as the rate at which the moving frame of reference moves downstream essentially determines how a spatial change in momentum appears as a temporal rate of change. What does this mean in mathematical terms? Consider the mean velocity, here constrained to vary only in time and distance downstream u = u x, t( )Find the derivative of velocity (time and space), using the chain rule: 412.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 The time rate of change in velocity is du dtdu dt=!u !tdtdt+!u !xdxdt, so divide through by dt: Note that dxdt! u , which explains how the material velocity plays a role: du dt=!u !t+ u !u !xwhere the first term on the RHS is the rate of change at a fixed point (the unsteady term), and the second is the rate of change associated with flow from one point to another (the convective acceleration term). Unsteady flow relates to a rising or falling hydrograph, the convective accelerations to flow around bends or over obstructions. 1D. Steady, Uniform Flow From above we have for conservation of momentum: hxhggxuutudtudb!""#$#""""##%%=&'()*++= sinSteady flow = no change of velocity in time at a fixed point => Uniform flow = no change of velocity in space at a fixed time => Under these conditions, the conservation of momentum reduces to: Re-writing we get: !"#$%&'=xhghb(()*+sin!!t= 0!!x= 0hxhggb!""#$#%%= sin0The term in brackets is approximately equal to the water surface slope. Alluvial channels usually have slopes < 5°, and the small angle approximation (sinα ~ tanα =-dz’/dx’ = So; where z’ is oriented in the vertical and denotes bed elevation) is often used. Thus basal shear stress is often approximated as (using S for water surface slope): ghSb!"=(τb also related to mean flow velocity and the shape of velocity profiles: next lecture) Note that