HY-8 and Inlet Control

HY-8's documentation on how it calcuates inlet control is amazingly lacking. Here is the inlet control equation selection flowchart, in its entirety:

Determine Applicable Inlet Control Equation

  1. IF circle or box with IMPROVED INLETS then use INLET equations.
  2. For Straight (previously called conventional) INLETS
    1. If Q is < Q at .5D, then assume LOW FLOW INLET CONTROL:
      1. calculate CRITICAL DEPTH (DCO)
      2. calculate Section Properties
      3. VH = (Q / AC)^2 / 64.4
      4. IH = DCO * LMULT + (1 + KELOW) * VH * VHCOEF
        1. IF no Depression THEN IHI = IH + I1E
        2. For Depression, HF = IH and check head on CREST.
    2. If Q > Q at .5D, but < Q at 3D, then use INLET REGRESSION EQUATIONS.
    3. If Q > Q at 3D, then assume HIGH FLOW INLET CONTROL.
      1. IH = (Q / CDAHI)^2 + .5 * RISE
      2. IF no Depression THEN IHI = IH + I1E
        1. For Depression, HF = IH and check head on CREST.

Straight Inlet Equations

  1. For IRREGULAR shape, X = Q / (AC * SQR(RISE))
    1. IF X <= .5 THEN IH = (A(1) * (X / .5)) * RISE
    2. ELSE IH = (A(J - 1) + (A(J) - A(J - 1)) * ((X - J + 2) / INC)) * RISE
  2. For all others shapes, X = Q / (SPAN * SQR(RISE^3)): SR = SR(IC)
    1. IH = (A + (B + (C + (D + (E + F * X) * X) * X) * X) * X - SR * S0) * RISE
  3. Headwater elevation (IHI) = IH + I1E if no Depression.
  4. For Depression, CREST headwater is checked.

So, a couple of things become readily apparent. There's a lot of recognizable equations in somewhat nonstandard forms, there's a lot of recognizable variables in somewhat nonstandard forms, and many of the variables are squished together to make them even more unrecognizable. I'd like to think that they've heard of Equation Editor over at FHWA, but this section gives me pause.  Let's see if we can't do better.

Unsubmerged Inlet Equations

The first formula (2.1.3) is relatively easy, since it's the standard velocity head equation. In place of the velocity, we simply divide discharge by the cross-sectional flow area assuming critical depth.

$$H_{V_c} = \frac{\left(Q / A_c \right)^2}{2 g}$$

The second equation (2.1.4), however, is a bit more odd. I'm still not sure what the LMULT and VHCOEF are supposed to be. Length multiplier? Velocity head coefficient? Where are these variables listed again? Looking at the differences between the computed results with and without the terms, it seems as though the LMULT term is 1 (at least for single barrel culverts), and the velocity head term can be estimated around 0.95.  So, simplifying HDS-5 Eq. A.1 should reveal something similar to this:

$$HW_U = y_c + 0.95 \left( 1 + K_e \right) H_{V_c} $$

This may not be correct, but the results I've gotten independently of HY-8 seem to indicate that they are "close enough."

A Brief Aside

Before moving on, a word on depths.  Since there's multiple formulas given to figure out the depths at a particular discharge, it seems odd that HY-8 decided to choose the formula based on the calculated flow depth.  So, the question is, which equation controls the flow depth for selecting a given equation?  Luckily, it seems as though the depths proceed from low to high.  If 2.1.4 is over half the opening height, then it goes to the regression equations.  Once the regression equations are over 3 times the inlet height, then the submerged equations are selected.

Transition Inlet Equations

This part of the HY-8 listed equations is relatively straightforward, and pretty much matches the guidance from HDS-5.  Stripping away the weird presentation (see above), the regression equation is exactly as it is in HDS-5 A.5.

$$HW_T = a + b \left[ \frac{Q}{B D^{3/2} } \right] + c \left[ \frac{Q}{B D^{3/2} } \right]^2 + d \left[ \frac{Q}{B D^{3/2} } \right]^3 + e \left[ \frac{Q}{B D^{3/2} } \right]^4 + f \left[ \frac{Q}{B D^{3/2} } \right]^5 + K_S S$$

So, a brief word on that last term.  HY-8 insists on the minus SR term times the pipe slope for the slope correction factor, whereas HDS-5 adds the slope correction factor and makes the Ks term negative for most pipe inlets.  I have no idea why the two publications feel the need to make this more more confusing than it should be.

Submerged Inlet Equations

Okay, everything so far is somewhat straightforward.  The submerged equations required a bit more digging.

This one throws out the following jumble of variables "CDAHI" and expects us to know exactly what that is without any further explanation.  A is typically area, HI could be Height of Inlet (but this directly conflicts with the IH term...).  CD is the orifice coefficient, which can be found by dividing 1 by the square root of 1 plus the ke inlet coefficient.  

$$HW_S =\left( 1 + k_e \right)  \left( \frac{Q}{A HI } \right)^2 + \frac{D}{2}$$

I'm still not quite sure what all is baked in that HI term.  Considering this is a variant of the orifice equation, I'm guessing there's a 2g term in there, but the numbers don't quite add up.  The term is generally consistent for all discharges for a particular culvert design, but I haven't been able to figure it out just yet.

Putting It All Together

If I were to re-write the excerpts from the HY-8 user manual, here's how I'd do it.  I'd have no problems if someone from FHWA used this for the next addition.

$$ HW = \begin{cases} \frac{HW}{D} \lt 0.5, & y_c + 0.95 \left( 1 + K_e \right) \frac{\left(Q / A_c \right)^2}{2 g} \\ 0.5 \le \frac{HW}{D} \lt 3.0, & a + b \left[ \frac{Q}{B D^{3/2} } \right] + c \left[ \frac{Q}{B D^{3/2} } \right]^2 + d \left[ \frac{Q}{B D^{3/2} } \right]^3 + e \left[ \frac{Q}{B D^{3/2} } \right]^4 + f \left[ \frac{Q}{B D^{3/2} } \right]^5 + K_S S\\ \frac{HW}{D} \ge 3.0, & \left( 1 + k_e \right) \left( \frac{Q}{A HI} \right)^2 + \frac{D}{2}\\ \end{cases}$$

Where:
A Cross sectional flow area;
Ac Cross sectional flow area at critical depth;
a, b, c, d, e, f Regression equation coefficients (see HY-8 Manual Tables 1-6, p. 55-66);
B Width of culvert opening;
c Submerged inlet equation coefficient (see HDS-5 Table A.1-A.6);
D Height of culvert opening;
g Acceleration of gravity;
HW Headwater above culvert entrance relative to the culvert inlet invert;
Ke Culvert entrance coefficient (see HDS-5 Table A.1-A.6);
KS Slope correction factor (0.7 for mitered inlets, -0.5 for all others);
Q Discharge;
S Slope of the culvert; and
yc Critical depth of flow in culvert.