ASCE HEC-RAS Seminar January 25, 2006 Session 1B Hydraulic Data and Fundamental Behavior Affected by Uncertainty

Quote Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe to describe. Why does the universe go to all the bother of existing?” Stephen Hawking

Topics of Session Review of Steady, Non-Uniform Flow Flow Profiles Energy Losses Selection of n-Values Understanding Variation Effects of Uncertainty in Loss Calcs Uncertainty in Section Geometry Importance of Thresholds of Behavior

Uniform (Normal) Conditions vs Non-Uniform Conditions In “normal” flow, the water surface is parallel to the bed slope and the EGL. This is not a normal occurrence. All flow profiles only approach normal depth asymptotically, and it can take a great distance for the depth to equal normal. In non-uniform flow the depth changes so the water surface changes; we need to predict the change.

Specific Energy of Flow Specific Energy, H is the flow energy measured W.R.T. the channel bottom: H = d + V 2 /2g For a wide channel, V =q/d, and so, H = d + q 2 /2gd 2 For a given flow then, q 2 /2g = d 2 (H-d)

Specific Energy Diagram

Critical Conditions Critical flow occurs when the Froude Number ( ) is exactly 1. This is the point the flow can have min. energy, and depends only on flow rate not on geometry or roughness. If flow depth is greater than critical depth it is sub-critical, if less it’s super-critical.

Direction of Information Transfer If flow is sub-critical (F r < 1) the flow depth is affected upstream. If the flow is super-critical (F r > 1) the flow depth is affected downstream. Toss a rock in the flow, if ripples move upstream against the flow it is sub-critical. The Flow Regime also affects the changes in depth caused by channel transitions.

Channel Transitions In tranquil flow, a bottom change up causes a depth reduction, a width decrease also causes a depth reduction. In rapid flow, a bottom change down causes a small depth decrease and a width increase also causes a depth decease. THIS BEHAVIOR IS COUNTER INTUITIVE! Make a quick sketch to see if the behavior is possible

Specific Energy Diagram E1E1 E2E2 ΔZΔZ

For Channels That are NOT Wide In any case where the width is less than about 10 times the depth, the use of the depth as the hydraulic radius is less accurate. In those cases use: V=Q/A Depth = A/Top Width ( the hydraulic depth) Thus the critical depth is determined from:

Transitions—Specific Energy Analysis Calculate q, F r and E Determine if Rapid or Tranquil Determine if Energy is increased or decreased Sketch Specific Energy Diagram

Transitions (Analysis) In general, transitions can be changes in width or changes in bottom elevation. The basis of the water surface response to a transition is the specific energy diagram. For a more advanced analysis, energy losses must be incorporated.

Width Change

Bottom Change & Width Change

Energy Analysis The energy (for initial analysis) is assumed to be constant (no losses). The energy equation provides:

Solution in Simple Case The equation allows calculation of depths, widths, velocities, energies or flow, depending on what is given. The definition of continuity, energy and elevation is often combined with the energy equation to get a solution.

Simple Example If the flow is 30 cfs in a 10 ft wide channel with a depth of 3 ft and the width changes to 6 ft at the same time as the bottom is raised one ft, what is the depth and change in WSEL?

Hydraulic Jump The only way flow can cross from super- critical to sub-critical regimes is through a hydraulic jump. The location of a jump is determined be the relationship of the sequent depth to the incoming flow depth.

Hydraulic jump The energy lost in a jump is large! Sequent depth is:

Specific Energy of Jump ysys yiyi

Differential Equation of Channel Flow By rearranging the Energy equation: We get :

The Gradually Varied Flow Profiles The profile depends on: The ratio of flow depth to normal depth The ratio of flow depth to critical depth The bed slope Sustaining Mild Steep Critical Non-Sustaining Adverse Horizontal

Flow Profiles Draw critical and normal depth on channel profile, number zones from outer zone Mild y n > y c M1, M2, M3 Steep y n < y c S1, S2, S3 Critical y n = y c C1, C2, C3 Horizontal no normal H2, H3 Adverse no normal A2, A3

The Differential Equation of Non- Uniform Flow solved by steps: For channels with regular geometry the profile is calculated by balancing the energy equation for an assumed water depth. Resulting is a calculated distance along the channel to the point where the assumed depth will occur. This is “Direct Step”.

Equations

Conditions Rectangular 20 ft. wide channel with slope of and an n-value of conveying 800 cfs, ends at an abrupt drop Y n =8.01 Y c =3.68 Y(x) 4y c.7y c M2

Calculations

Numerical Sensitivity? What id the steps in Water depth were 0.5 ft? 0.01 ft? How close is close enough?

Roughness Estimates For lined channels the theoretical description of flow behavior is useful. Manning’s n-values, f, C and C H can be used. There is little advantage to not using n- values but f and C are more fundamentally correct

Roughness It it essential to recognize that open channel flow has variable flow geometry rather than only variable velocity (as in a pipe). Thus, the relative roughness (k s /D) changes. As the roughness changes so does the n-value (f and C also).

Roughness Roughness has components that are considered separately: 2-28 River Engineering Channel material Vegetation Alignment Channel Irregularity Channel Variation

Roughness Factors where: no = Base value for straight uniform channels n1 = Additive value due to cross-section irregularity n2 = Additive value due to variations of the channel n3 = Additive value due to obstructions n4 = Additive value due to vegetation m5 = Mulitiplication factor due to sinuosity

Channel Roughness Catalogs “Rules of Thumb ” Textural catalogs Photographic Catalogs of measured roughness values USGS Barnes REMEMBER the boundary roughness can change from bed-form changes induced by the flow.

Barnes

Barnes 2

Barnes 3

Dealing With Roughness Uncertainty Some Engineers have told me “n-value you pick doesn’t matter, nobody knows the correct number”… WTF, over?????? I use a range of reasonable values to calculate how the variables you are examining change with roughness changes. Does your decision change?? What is Sensitivity of your situation to the uncertainty?

Compound Channels In most real channels the change of channel area as the depth changes is not a smooth function. There are frequently floodplains where width increases enormously with a small change in depth. These situations are called compound channels.

Channel Geometry

Analysis

Sub-sections of the channel are identified with a zone of equal n-value. Water-to-water shear is neglected. Energy slope is the same for all zones. Sum of sub-section discharge is total discharge.

Alternative Analysis Would “average” n-value over entire channel be acceptable? When ? Why? What is influence of neglecting the water-to-water shear?? What is a practical limit to sub-section division?

Uncertainty in Simple Case

For Channels that Are NOT Regular – Standard Step. For channels that are irregular, the cross- sections are located at given positions. Therefore, a guess is made of the water level at the next section. Based on that guess the energy loss is calculated, the calculated water level is then compared to the guess, and the guess updated until an acceptable ‘closure’ at that section is obtained.

Standard Step Data

Standard Step Calculations All irregular channels require the use of standard step. Because the calculations of the energy loss is tedious the method is best computerized. There are many issues to consider in the calculation scheme. TOTAL Head Loss Average Roughness in Each location and between sections

Standard Step Equations

Standard Step Calculation Procedure Beginning at known conditions, guess Y 2, with channel shape calculate V 2, then S 2, and solve for what Y 2 satisfies the original energy equation. If guess and calculated value are the “same”, that is “correct” answer. Otherwise guess again.

The Limitations of Standard Step Method 1. Gradually Varied because hydrostatic Pressure is assumed 2. One-Dimensional 3. Steady because no time term is present 4. Small channel Slope (10%-20%) because y and H are assumed collinear.

Homework Look at the following publications in the references CD: The origin and Derivation of Ia/S in the Runoff Curve Number System NEH Part 630 Hydrology and HEH 4 (old) HEC-HMS Documentation Basic Hydraulic Principles River Engineering