[Paper Review] The Flowing System Gasdynamics. Part 3: Saint-Venant - Wantzel formula modern form

S. L. Arsenjev, I. B. Lozovitski+1 more|ArXiv.org|Feb 12, 2003

Aquatic and Environmental Studies3 citations

TL;DR

This paper presents a modernized form of the Saint-Venant–Wantzel formula for gas outflow velocity in flowing systems by integrating contact interaction via the static head law, eliminating the need for velocity and discharge coefficients and assuming a polytropic process. The reformulated equation establishes a unified spatial-energy relationship, enabling accurate computation of gas stream parameters in compressible flow systems with improved physical consistency and computational efficiency.

ABSTRACT

The modern form of the Saint-Venant - Wantzel formula for an outflow velocity of gas stream from flowing element is submitted. Taking into account of contact interaction of gas stream with the streamline surface in the form of the static head law has allowed to find the spatial-energy liaison in flowing system. The physically correct combination of mechanics of contact interaction and thermodynamics of fluid medium in one formula has allowed simultaneously to be liberated from the velocity coefficient and the discharge coefficient and polytropic process. In the new form the formula has gained the key character for computation of parameters of motion and state of gas stream in the flowing system.

Motivation & Objective

• To develop a modern, physically consistent form of the Saint-Venant–Wantzel formula for gas outflow velocity in flowing systems.
• To eliminate reliance on empirical coefficients such as velocity and discharge coefficients in gas dynamics calculations.
• To unify the mechanics of contact interaction with thermodynamic behavior of compressible fluids in a single analytical framework.
• To establish a spatial-energy linkage in flowing gas systems through a revised energy balance incorporating surface interaction effects.
• To enable more accurate and direct computation of gas stream motion and state parameters without assuming polytropic processes a priori.

Proposed method

• Incorporates the static head law to model contact interaction between the gas stream and the streamline surface.
• Reformulates the Saint-Venant–Wantzel formula by embedding the static head effect directly into the energy balance.
• Integrates fluid mechanics and thermodynamics into a single equation, removing the need for separate coefficients.
• Applies the reformulated equation to compute velocity, pressure, and temperature distributions in flowing gas systems.
• Derives a unified expression that inherently accounts for energy transfer and surface interaction in compressible flows.
• Validates the new form through theoretical consistency and physical coherence, avoiding ad hoc assumptions.

Experimental results

Research questions

• RQ1 How can the Saint-Venant–Wantzel formula be reformulated to eliminate empirical coefficients like velocity and discharge coefficients?
• RQ2 What is the role of contact interaction with the streamline surface in shaping the energy distribution of a gas stream?
• RQ3 Can a single equation simultaneously describe mechanical contact effects and thermodynamic behavior in compressible gas flows?
• RQ4 How does the inclusion of the static head law improve the physical accuracy of gas outflow predictions?
• RQ5 What is the resulting spatial-energy relationship in a flowing gas system under the new formulation?

Key findings

• The new formula eliminates the need for velocity and discharge coefficients by embedding their effects through the static head law.
• The reformulated equation provides a physically consistent link between spatial geometry and energy distribution in flowing gas systems.
• The integration of contact mechanics and thermodynamics into one equation removes the assumption of a predefined polytropic process.
• The resulting expression enables direct computation of gas stream parameters such as velocity, pressure, and temperature with improved accuracy.
• The formula achieves a unified description of flow behavior that is both mathematically and physically coherent.
• The modern form demonstrates enhanced applicability for computational modeling of compressible flows in engineering systems.