Specific Speed and Design Guidelines for Turbomachinery
Theory and Physics
Overview
Specific speed is a parameter used for selecting the type of turbomachinery, right?
Yes. Specific speed is a dimensionless parameter calculated from flow rate, head (pressure difference), and rotational speed. It is the most fundamental indicator for determining the optimal machine type.
Definition of Specific Speed
Please show me the formula.
This form is widely used in Japan.
$N$: Rotational speed [rpm], $Q$: Flow rate [$m^3/min$], $H$: Head [m]. Although not dimensionless, this dimensional form is the most commonly used in practice.
Internationally, the dimensionless specific speed ($\Omega_s$ or $\omega_s$) is also used.
How does the specific speed value determine the type?
| Specific Speed $N_s$ (Dimensional) | Type | Characteristics |
|---|---|---|
| 100～300 | Centrifugal (Radial) | High head, low flow rate |
| 300～600 | Mixed Flow | Medium head, medium flow rate |
| 600～1500 | Axial Flow | Low head, high flow rate |
Cordier Diagram
What is the Cordier diagram?
It's a diagram showing the optimum line for dimensionless specific speed $\omega_s$ and dimensionless specific diameter $\delta_s$.
The relationship between $\omega_s$ and $\delta_s$ that achieves optimum efficiency is drawn as the Cordier line, which holds true for both pumps and turbines. As a starting point for design, you first check if your point lies on the Cordier line.
So you check with the Cordier diagram before starting CFD, right?
Yes. Determining initial values for basic dimensions (diameter, blade width) from the specific speed and Cordier line before proceeding to CFD is an efficient workflow.
History of the Specific Speed Concept – A Design Similarity Rule Born from 19th-Century Hydraulic Engineers
The concept of Specific Speed (Ns) was empirically developed by hydraulic machinery engineers in the late 19th century to compare similar water turbines. Italian hydraulic engineers and Germany's Vogel independently proposed similar parameters, and a unified definition was established in Europe and America in the 1910s-20s. Initially used as "hydraulic specific speed" for turbine design, it was later extended to centrifugal pumps and compressors. The dimensionless form (Shape Number Omega = omega*sqrt(Q)/(g*H)^(3/4)) was established in the 1940s and later to clarify its physical meaning. In Japanese hydraulic machinery engineering, the domestic calculation formula for Ns (Ns = N*sqrt(Q)/H^(3/4)) is still used today. Since its coefficient differs from the ISO dimensionless formula, confusion can easily arise when comparing with international papers—it is a basic practice to clearly state which definition is being used.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, it becomes a steady flow, right? This term describes that "period of change." The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since computational cost is significantly reduced, solving first with a steady-state approach is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while, they naturally mix. That's molecular diffusion. Now a question—honey or water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, confusion between absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result, like turning on a heater in a winter room but the warm air doesn't rise.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path of molecules ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock wave capturing required), free surface flow (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³@20°C, Water: approx. 998 kg/m³@20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
1D Design and CFD Integration
After deciding the type with specific speed, what is the design flow to CFD?
Let me show a typical design flow.
1. Specification Determination: Flow rate Q, Head H (or pressure ratio), Rotational speed N
2. Specific Speed Calculation: Calculate $N_s$ and select the type
3. Mean-Line Design: Determine velocity triangles, blade angles, meridional dimensions
4. Throughflow Analysis: Calculate 2D meridional flow field
5. 3D Blade Shape Definition: Generate 3D blade surfaces using BladeGen, etc.
6. 3D CFD: Detailed analysis with TurboGrid + CFX
7. Optimization: Parametric study or automatic optimization
What do you do in Mean-Line design?
Design the velocity triangle at the mid-span. Determine blade angles from inlet/outlet absolute velocity, relative velocity, and blade speed, and check load validity using the de Haller number or diffusion factor.
Throughflow Analysis
What is Throughflow analysis?
It's a 2D analysis that solves the flow on the meridional plane (r-z plane) assuming axisymmetry. The influence of the blade row is simulated as a body force representing blade force. Since you obtain the spanwise velocity and pressure distribution, it forms the basis for determining the blade angle distribution from hub to tip.
| Tool | Method | Developer |
|---|---|---|
| Concepts NREC COMPAL/AXIAL | Streamline Curvature Method | Concepts NREC |
| AxSTREAM (SoftInWay) | Streamline Curvature Method | SoftInWay |
| NUMECA AutoBlade | Automatic Blade Definition | NUMECA |
| Vista CCD (Ansys) | 1D Centrifugal Design | Ansys |
What is the Streamline Curvature Method?
It's a method that calculates the spanwise pressure gradient based on the curvature of streamlines on the meridional plane. It's the standard method for Throughflow analysis of turbomachinery. Since results are obtained in seconds, it's optimal for parametric design.
Cross-checking Specific Speed and CFD Results – Relationship between Analytical Verification of η-Ns Curves and Numerical Calculation Accuracy
Specific Speed Ns is a design similarity parameter that non-dimensionalizes turbomachinery by flow rate, head, and rotational speed. The optimal blade type is determined from the Ns value. After CFD analysis, before comparing efficiency with experimental values, it's important to first cross-check with known Ns-η (specific speed-efficiency) charts. If the result deviates from the maximum efficiency Ns-η curve compiled from experimental collections like Lomakin(1958) or Kaplan(1935), there is either a design problem or a CFD model problem. Especially if "CFD efficiency exceeds the upper limit of the Ns-η curve (physically impossible high efficiency)," it's almost certainly due to missing loss modeling (volute loss, bearing loss, etc.) or erroneous boundary condition settings.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Scheme (MUSCL, QUICK, etc.)
Suppresses numerical oscillations while maintaining high accuracy using limiter functions. Effective for capturing shock waves or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for continuity, momentum, and energy equations drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2~0.3, Velocity: 0.5~0.7. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Guideline for internal iteration count: 5~20 times. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it's a discretization method that reflects the physics that upstream information determines downstream. Although it's first-order accurate, it is highly stable because it correctly captures the flow direction.
Practical Guide
CFD Considerations by Specific Speed
Does the CFD approach change depending on specific speed?
It changes significantly. Let's summarize the characteristics for each type.
| Specific Speed Range | Type | Main CFD Challenges | Recommended Meshing Method |
|---|---|---|---|
| Low (100-200) | Centrifugal (Low Flow) | Diffuser loss, Recirculation | TurboGrid + Volute Unstructured |
| Medium-Low (200-400) | Centrifugal (Standard) | Jet/Wake Structure | TurboGrid |
| Medium (400-600) | Mixed Flow | Effect of Meridional Curvature | TurboGrid (Axial-Radial Mixed) |
| High (600-1000) | Axial Flow (Large Hub Ratio) | Tip Leakage, Secondary Flow | TurboGrid |
| Very High (>1000) | Axial Flow (Small Hub Ratio) | Incompressible Flow, Stall | TurboGrid or Unstructured |
Is CFD for mixed-flow pumps particularly difficult?
Because the meridional curvature is large, the mesh in the inter-blade flow passage is prone to distortion. It's important to select J-type or L-type topologies in TurboGrid and set them to follow the meridional curvature.
Relationship Between Specific Speed and Efficiency
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