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In this paper we provide an explicit description of normal almost contactstructures obtained from Cartan-Ehresmann connections (gauge fields) onprincipal $S^{1}$-bundles over complex flag manifolds. The main feature of ourapproach is to employ elements of representation theory of complex simple Liealgebras in order to describe and classify these structures. We use thesenormal almost contact structures to explicitly describe a huge class of compactHermitian non-K\"{a}hler manifolds obtained from products of principal$S^{1}$-bundles over complex flag manifolds. Moreover, we obtain from ourdescription several concrete examples of 1-parametric families of complexstructures on products of principal $S^{1}$-bundles over flag manifolds, theseconcrete examples generalize the Calabi-Eckmann manifolds. Further, as anapplication of our main results in the setting of KT structures on toricbundles over flag manifolds, we classify a huge class of explicit examples ofCalabi-Yau structures with torsion (CYT) on certain Vaisman manifolds(generalized Hopf manifolds). Also as an application of our main results, weprovide several new concrete examples of astheno-K\"{a}hler structures onproducts of compact homogeneous Sasaki manifolds.

Hermitian non-K\"{a}hler structures on products of principal  $S^{1}$-bundles over complex flag manifolds and applications in Hermitian  geometry with torsion