End Sills In Stilling Basins: USBR Solid vs Dentated Design & Solved Examples (2026)

1. Terminal Hydraulic Structures and Downstream Riverbed Protection

Discharging floodwaters from dam spillways and outlet conduits into natural alluvial river channels requires careful transition management. The exit flow leaving a stilling basin often retains residual kinetic energy and high bottom velocities capable of undermining the downstream toe of concrete aprons. Constructing engineered end sills in stilling basins prevents bed degradation and anchors the hydraulic jump.

                      TERMINAL SILL & GROUND ROLLER FLOW
        Stilling Basin Floor
        ======================\            Upward Deflected Surface Current
                               \          /=================================
                                [End Sill]
        =============================|     <---- Reverse Ground Roller
                                     |           (Deposits Sediment at Toe)
                                     +--------------------------------------
                                              Alluvial Riverbed

Located at the downstream terminus of the concrete apron, an end sill acts as a terminal deflector. By lifting high-velocity bottom currents toward the water surface, the sill transforms aggressive boundary shear stresses into benign surface turbulence, inducing a reverse ground roller that actively deposits bed material against the apron cutoff wall.

Open Channel Flow Fundamentals and Energy Dissipation

2. Fluid Mechanics of End Sills: Jet Deflection and Ground Roller Formation

The primary hydrodynamic mechanism of an end sill involves vertical momentum redirection:

1. Upward Jet Deflection: High-velocity bed currents strike the upstream face of the sill and are forced upward toward the free surface. This creates a surface boil where residual kinetic energy dissipates into atmospheric turbulence.
2. Reverse Ground Roller Generation: The upward trajectory of the primary jet creates a localized low-pressure zone directly downstream of the sill. This pressure gradient draws bottom water upstream, establishing a clockwise ground roller.
3. Sediment Scour Mitigation: Instead of sweeping alluvial sands and gravels downstream, the ground roller pulls riverbed sediment upstream, banking it against the apron cutoff wall to prevent structural undermining.

                         GROUND ROLLER RECIRCULATION
                                Deflected Jet (Surface Boil)
                                     ===========>
                                    /
                     [ End Sill ]  /      Low Pressure Eddy Zone
                     +----------+ /
                     |          |/   <---- Reverse Bed Current
        =============+==========+=====================================
                      Cutoff Wall          Accumulated Bed Material

Without an end sill, the bottom jet exits horizontally along the riverbed, creating a deep scour hole directly adjacent to the concrete slab that causes cutoff wall undermining and progressive slab collapse.

Baffle Block Hydrodynamics and Apron Appurtenances

3. USBR Solid End Sills: Geometry, Performance, and Applications

The United States Bureau of Reclamation (USBR) standardized solid end sills for low-to-intermediate head structures (USBR Type III and Type IV stilling basins):

                        USBR SOLID SLOPING END SILL
                                     Top Flat
                                      +----+
                         Sloping Face |    | Vertical Downstream Face
                           (2:1 H:V)  |    | (or 1:1 Chamfer)
                                     /|    |
                         ===========+======+======================
                                    |< h4 >| (Height h4 = 0.06 - 0.20 y2)

Dimensional Standards for Solid End Sills:

• Height ($h_4$): Typically sized as a fraction of theoretical conjugate depth: $h_4 \approx 0.06 y_2 \text{ to } 0.20 y_2$.

• Upstream Slope: A $2:1$ (Horizontal : Vertical) sloping face is standard for USBR Type III basins to minimize dynamic impact shocks and cavitation inception while effectively deflecting bottom currents.

• Top Width: $w_{top} \approx 0.20 h_4 \text{ to } 0.50 h_4$.

• Downstream Face: Vertical or stepped with a cutoff wall extending into non-erodible strata.

Solid sills are cost-effective, straightforward to construct, and structurally durable in rivers with heavy bedload transport where rocks and cobbles would damage dentated teeth.

4. USBR Dentated End Sills: Geometry, Slot Hydraulics, and Energy Dissipation

For high-velocity, high-head spillway structures ($v_1 > 15 \text{ to } 18 \, \text{m/s}$), solid sills can cause excessive upward jet throw that disturbs downstream banks. USBR Type II stilling basins incorporate dentated end sills featuring alternating teeth and open slots.

                        USBR DENTATED END SILL GEOMETRY
                       |<--- w4 --->|<--- s4 --->|
                       +------------+            +------------+  -+-
                       |   Tooth    |    Slot    |   Tooth    |   | h4
                       |  (Dentate) |   (Space)  |  (Dentate) |   |
                       +------------+============+------------+  -+-
                       |<------------- Basin Width B ------------->|

Dimensional Standards for USBR Type II Dentated Sills:

• Height ($h_4$): Sized as $h_4 = 0.20 y_2$.

• Tooth Width ($w_4$): Sized equal to tooth height: $w_4 = 0.15 y_2 \text{ to } 0.20 y_2$.

• Clear Slot Spacing ($s_4$): Sized equal to tooth width: $s_4 = w_4 = 0.15 y_2 \text{ to } 0.20 y_2$.

• Upstream Slope: Continuous $2:1$ sloping ramp extending from the basin apron to the top of the teeth.

Slot Hydraulics and Energy Dissipation:

Dentated sills split the exiting flow into two distinct streams:
1. Upper Stream: Water flowing over the teeth is deflected upward toward the surface.
2. Lower Stream: Water passing through the open slots maintains a low-velocity bottom current that cushions the riverbed.

The interaction between these two interlocked streams generates 3D mixing and shear layers, smoothing the downstream velocity profile and reducing water surface waves.

5. Comparative Analysis: Solid vs Dentated End Sills

6. Scour Control and Riprap Apron Interface Dynamics

The exit channel directly downstream of an end sill requires an engineered stone riprap transition to withstand residual shear stresses $\tau_0 = \gamma R_h S_f$.

                       RIPRAP TRANSITION DOWNSTREAM OF SILL
                 End Sill
                 +-----+
                 |     | \
        =========+=====+--\============================== Cutoff Wall
                           \  Layer of Graded Riprap (D50)
                            \~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                             \=========================== Filter Fabric

Riprap Sizing Criteria (USACE EM 1110-2-1601):

The median rock size $D_{50}$ for riprap aprons downstream of stilling basin end sills is calculated from exit velocity $v_{exit}$:

$$D_{50} = C_{scour} \cdot y_{exit} \cdot \left[ \frac{v_{exit}}{\sqrt{(S_s – 1) g y_{exit}}} \right]^{2.5}$$

Where $S_s \approx 2.65$ is stone specific gravity, $C_{scour} \approx 0.30 \text{ to } 0.38$ is a turbulence coefficient, and $y_{exit} \approx y_2$.

The riprap layer thickness is specified as $t_{riprap} \ge 1.5 D_{50}$ to $2.0 D_{50}$, placed over a geotextile filter fabric or graded gravel filter layer to prevent underlying fine soil migration.

7. Solved Numerical Example 1: USBR Type III Solid Sloping End Sill Design

Problem Statement

A rectangular stilling basin ($B = 14.0 \, \text{ m}$, $Q = 112.0 \, \text{ m}^3/\text{ s}$) for an intermediate-head dam has an upstream supercritical depth $y_1 = 0.75 \, \text{ m}$ and approach velocity $v_1 = 10.67 \, \text{m/s}$.

Determine:
1. Upstream Froude number $Fr_1$ and theoretical conjugate depth $y_2$.
2. Recommended height $h_4$, base width, and upstream slope for a USBR Type III solid end sill.
3. Basin length $L_b$ using USBR Type III empirical rules.

                 USBR TYPE III SOLID SILL SIZING
              B = 14.0 m, q = 8.0 m^2/s, y1 = 0.75 m
              ====================================\
                                                   \  Solid Sill (h4 = 0.60 m)
                                                    \    +---+
              =======================================+===+===+==========
              |<-------------- Lb = 8.9 m ------------->|

Step-by-Step Solution

Step 1: Hydraulic Parameters
$$q = \frac{Q}{B} = \frac{112.0}{14.0} = 8.0 \, \text{ m}^2/\text{ s}$$

$$Fr_1 = \frac{v_1}{\sqrt{g y_1}} = \frac{10.67}{\sqrt{9.81 \times 0.75}} = \frac{10.67}{2.7125} = 3.933 \approx 4.0$$

Theoretical conjugate depth $y_2$:
$$y_2 = \frac{y_1}{2} \left( \sqrt{1 + 8 Fr_1^2} – 1 \right) = \frac{0.75}{2} \left( \sqrt{1 + 8(3.933)^2} – 1 \right)$$

$$y_2 = 0.375 \times (\sqrt{1 + 123.75} – 1) = 0.375 \times (11.169 – 1) = 0.375 \times 10.169 = 3.813 \, \text{ m}$$

Step 2: Sizing the Solid Sloping End Sill
– Solid End Sill Height ($h_4$):
$$h_4 = 0.15 y_2 = 0.15 \times 3.813 = 0.572 \, \text{ m} \implies \text{Adopt } h_4 = 0.60 \, \text{ m}$$

• Upstream Face Slope: $2:1$ (Horizontal : Vertical) $\implies$ Horizontal run $L_{ramp} = 2 \times 0.60 = 1.20 \, \text{ m}$.

• Top Crest Width: $w_{top} = 0.30 \, \text{ m}$.

• Downstream Vertical Cutoff Wall: Depth $d_{cutoff} = 1.50 \, \text{ m}$ below riverbed.

Step 3: Total Basin Length ($L_b$)
For USBR Type III basin:
$$L_b = 2.7 y_2 = 2.7 \times 3.813 = 10.30 \, \text{ m}$$

8. Solved Numerical Example 2: USBR Type II Dentated End Sill Sizing for High-Head Spillway

Problem Statement

A high-head spillway chute delivers design flood flow $Q = 480 \, \text{ m}^3/\text{ s}$ across width $B = 24.0 \, \text{ m}$. Supercritical flow parameters at the stilling basin toe are $y_1 = 0.90 \, \text{ m}$ and $v_1 = 22.22 \, \text{m/s}$ ($Fr_1 = 7.48$). Because $v_1 > 15 \, \text{m/s}$, a USBR Type II basin with a dentated end sill is specified.

Calculate:
1. Sequent depth $y_2$.
2. Dentated sill height $h_4$, tooth width $w_4$, and clear slot spacing $s_4$.
3. Total number of teeth $N_{teeth}$ and slots across basin width $B$.
4. Total USBR Type II basin length $L_b$.

Step-by-Step Solution

Step 1: Sequent Conjugate Depth ($y_2$)
$$y_2 = \frac{0.90}{2} \left( \sqrt{1 + 8(7.48)^2} – 1 \right) = 0.45 \times (\sqrt{1 + 447.60} – 1) = 0.45 \times (21.18 – 1) = 9.081 \, \text{ m}$$

Step 2: Dentated End Sill Dimensions (USBR Type II Rules)
– Sill Height:
$$h_4 = 0.20 y_2 = 0.20 \times 9.081 = 1.816 \, \text{ m} \implies \text{Adopt } h_4 = 1.80 \, \text{ m}$$

• Tooth Width ($w_4$):
  $$w_4 = 0.15 y_2 = 0.15 \times 9.081 = 1.362 \, \text{ m} \implies \text{Adopt } w_4 = 1.35 \, \text{ m}$$

• Clear Slot Spacing ($s_4$):
  $$s_4 = w_4 = 1.35 \, \text{ m}$$

• Upstream Ramp Slope: $2:1 \implies L_{ramp} = 2 \times 1.80 = 3.60 \, \text{ m}$.

Step 3: Number of Teeth and Layout
$$N_{teeth} = \frac{B}{w_4 + s_4} = \frac{24.0}{1.35 + 1.35} = \frac{24.0}{2.70} = 8.89 \implies 9 \text{ teeth}$$
Layout: 9 teeth of width $1.33 \, \text{ m}$ separated by 8 interior slots of width $1.35 \, \text{ m}$ and two $0.60 \, \text{ m}$ end half-slots adjacent to sidewalls.

Step 4: Basin Length ($L_b$)
For USBR Type II basins ($L_b \approx 4.0 \text{ to } 4.3 y_2$):
$$L_b = 4.3 y_2 = 4.3 \times 9.081 = 39.05 \, \text{ m}$$

9. Solved Numerical Example 3: Hydrodynamic Thrust and Overturning Moment on End Sill

Problem Statement

A solid end sill of height $h_4 = 0.80 \, \text{ m}$ spans channel width $B = 10.0 \, \text{ m}$. Flow arrives at the sill with subcritical velocity $v = 2.50 \, \text{m/s}$ and total flow depth $y = 4.0 \, \text{ m}$.

Calculate the total horizontal hydrodynamic force $F_h$ (hydrostatic differential plus dynamic stagnation thrust) and resulting base overturning moment $M_{OT}$.

                 FORCES ACTING ON SOLID END SILL
                     v = 2.5 m/s
                     y = 4.0 m
                     =======>   +-------+
                                |  h4 = | F_dynamic
                     P_upstream | 0.80 m| <--- P_downstream
                     ==========+=======+========================

Step-by-Step Solution

Step 1: Dynamic Stagnation Pressure Force ($F_{dyn}$)
Frontal projected area:
$$A_{front} = B \times h_4 = 10.0 \times 0.80 = 8.0 \, \text{ m}^2$$

Dynamic force:
$$F_{dyn} = C_D A_{front} \frac{\rho v^2}{2} = 1.10 \times 8.0 \times \frac{1000 \times (2.50)^2}{2} = 8.8 \times 3125 = 27,500 \, \text{ N} = 27.50 \, \text{ kN}$$

Step 2: Hydrostatic Differential Force ($\Delta P$)
– Upstream water depth on sill: $y_{up} = 4.0 \, \text{ m}$

• Downstream tailwater depth: $y_{down} = 4.0 – 0.80 = 3.20 \, \text{ m}$

• Upstream hydrostatic force on sill height $h_4$:
  $$P_{up} = \gamma B \left( y_{up} – \frac{h_4}{2} \right) h_4 = 9.81 \times 10.0 \times (4.0 – 0.40) \times 0.80 = 98.1 \times 3.60 \times 0.80 = 282.53 \, \text{ kN}$$

• Downstream hydrostatic force on sill:
  $$P_{down} = \frac{1}{2} \gamma B y_{down}^2 \text{ (on bottom 0.80 m)} = 9.81 \times 10.0 \times \left( \frac{0.80}{2} \right) \times 0.80 = 31.39 \, \text{ kN}$$

• Net Hydrostatic Force:
  $$\Delta P = P_{up} – P_{down} = 282.53 – 31.39 = 251.14 \, \text{ kN}$$

Step 3: Total Horizontal Design Force ($F_h$)
$$F_h = F_{dyn} + \Delta P = 27.50 + 251.14 = 278.64 \, \text{ kN}$$

Step 4: Overturning Moment ($M_{OT}$)
Assuming resultant acts at height $\bar{y} \approx 0.45 h_4 = 0.45(0.80) = 0.36 \, \text{ m}$:

$$M_{OT} = F_h \times \bar{y} = 278.64 \, \text{ kN} \times 0.36 \, \text{ m} = 100.31 \, \text{ kN}\cdot\text{ m}$$

10. Construction Specifications and Armoring Standards: USBR and USACE Guidelines

1. Concrete Abrasion Resistance (USACE EM 1110-2-2102):
   – End sills encounter continuous sediment abrasion from reverse ground rollers. Concrete specifications require compressive strength $f’_c \ge 35 \, \text{ MPa}$ with hard quartzite or basalt coarse aggregates (Los Angeles Abrasion loss $< 20\%$).
2. Deep Cutoff Walls (USBR Design Standards No. 14):
   – Monolithic cutoff walls extending below the sill base must penetrate at least $1.5 \text{ to } 2.5 \, \text{ m}$ into native bedrock or sound alluvium to prevent piping seepage beneath the apron slab.
3. Subsurface Relief Drains:
   – Underdrain perforated collector pipes with non-return flap valves installed immediately upstream of the end sill relieve excessive uplift pore-water pressures during rapid tailwater drawdown.

Water Surface Profile Computations and Tailwater Control

11. End Sills in Stilling Basins Engineering Synthesis

Proper engineering of end sills in stilling basins ensures a stable transition between turbulent energy dissipation structures and natural river ecosystems. Whether selecting solid sloping sills for moderate-head chutes or dentated sills for high-velocity spillways, civil engineers utilize vertical momentum redirection to suppress downstream bed scour, preserving structural integrity across water resource infrastructure.

References & Standards Cited

1. US Department of the Interior, Bureau of Reclamation (USBR). (1987). Hydraulic Design of Stilling Basins and Energy Dissipators (Engineering Monograph No. 25). Denver, CO: USBR.
2. US Army Corps of Engineers (USACE). (1991). Hydraulic Design of Flood Control Channels (Engineer Manual EM 1110-2-1601). Washington, DC: Department of the Army.
3. Peterka, A. J. (1984). Hydraulic Design of Stilling Basins and Energy Dissipators. USBR Engineering Monograph No. 25, 8th printing.
4. Chow, V. T. (1959). Open-Channel Hydraulics. New York: McGraw-Hill.
5. Mason, P. J. (1982). The Choice of Hydraulic Energy Dissipator for Dam Spillways. Proceedings of the Institution of Civil Engineers, 72(2), 209-222.

Frequently Asked Questions (FAQ)

What is the main purpose of an end sill in a stilling basin?

The main purpose is to deflect high-velocity bottom currents upward toward the surface, generating a reverse ground roller that deposits alluvial bed material against the apron cutoff wall to prevent riverbed scour.

When should a dentated end sill be chosen over a solid end sill?

A dentated end sill is chosen for high-head, high-velocity spillways ($v_1 > 15 , text{m/s}$, USBR Type II basins) where a solid sill would throw an excessively high jet that disturbs downstream riverbanks.

How high should a USBR Type III solid end sill be?

USBR Type III guidelines specify a solid end sill height of $h_4 approx 0.06 y_2 text{ to } 0.20 y_2$ (typically $0.15 y_2$), with an upstream sloping face inclined at $2:1$ (Horizontal:Vertical).

Why does an end sill create a reverse ground roller?

Lifting the main jet toward the surface creates a localized low-pressure zone directly downstream of the sill. Downstream bed water flows upstream into this low-pressure void, establishing a recirculating ground vortex.

What maintenance challenges are associated with dentated end sills?

Dentated end sills can trap floating logs, woody debris, and river boulders inside the slots. In cold regions, ice impact can cause spalling on vertical tooth edges, requiring reinforced chamfers and regular inspection.