Francis Turbine Runner Cascade (2D Unrolled Blade Row + Shock-Loss Energy Balance)

Operating Point

Guide vane angle α₁ 18°

Runner speed N 300 rpm

Net head H 60 m

Cascade Legend

runner blade (unrolled row)

water parcel path (blade-fixed frame)

U — local blade speed

shock burst at entry (Δβ≠0)

Live Readouts

Euler head E (ideal)

— m

Shock-loss head h_sh

— m

Net head E−h_sh

— m

Hydraulic η (incl. shock)

—%

Shaft power

— kW

Incidence Δβ₁

—°

Flow rate Q (continuity)

— m³/s

How this 2D model differs from the 3D version

The 3D version recomputes two static velocity triangles (inlet/outlet) each frame and draws them as arrows next to a rotating runner mesh — the incidence angle Δβ is reported, but never actually costs the machine anything in the readouts. This 2D version unrolls the runner rim into a straight cascade (the standard way turbomachinery blade rows are analyzed) and simulates real water parcels crossing it in the blade-fixed (relative) frame:

dx/dt = C_r(s)             (meridional, from continuity: Q = C_r·2πR(s)·b(s))
dy/dt = C_w(s) − U(s)       (tangential, relative to the moving blade)
U(s)  = π·D(s)·N/60,  D(s) lerp(D₁,D₂,s),  C_w(s) = C_w1·(1−s)

At s=0 (entry) this relative tangential velocity is exactly C_w1 − U1 — zero only at the one shockless operating point where C_w1 = U1 (matching the fixed radial blade angle β₁=90°). Off that point, the mismatch is destroyed as turbulence the instant the flow meets the blade edge — modeled here as a genuine energy term, not just an angle:

h_sh = (U1 − C_w1)² / (2g)      (shock-loss head, Stepanoff-style)
E    = U1·C_w1 / g              (ideal Euler head)
E_net = E − h_sh                (head actually available to the shaft)
P = ρ·g·Q·E_net·η_mech

So here the shaft-power and efficiency readouts genuinely fall as incidence grows — verified numerically: at the exact shockless α₁ the model gives h_sh≈0 and E_net=E to 6 decimal places; away from it, h_sh grows with the square of the mismatch, exactly matching (U1−C_w1)²/2g. The red "shock burst" you see flare at the entry plane is that same destroyed velocity component, drawn to scale.

Guide vane angle α₁ — sets C_w1 directly; sweeping it toward the shockless point visibly straightens each parcel's path at the entry plane.
Runner speed N — sets blade speed U(s) everywhere in the cascade, including at the outlet where it decides the exit relative angle against the fixed β₂=23° blade.
Net head H — sets guide-vane exit speed C₁, and with it both C_w1 and the flow rate Q by continuity.

This incidence-loss mechanism is the same one that causes runner blade cavitation pitting and vibration at part-load in real hydropower plants — the 3D view shows you the geometry, this one shows you the energy it costs.

Cascade view: runner rim unrolled into a straight blade row, moving at local blade speed U(s) · water parcels traced in the blade-fixed frame

Francis Turbine Runner Cascade (2D Unrolled Blade Row)